{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3sin(3\u03c0(2|(-3x - 4) - 2| + 1)) - 3) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|(1sin(1\u03c0(5x + 1)))| + 1)^2 + 3(1|(1sin(1\u03c0(5x + 1)))| + 1) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-3sin(1\u03c0(1((1x - 1)/(-2x - 3)) - 2) + 1) - 3) - 2| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2(1sin(2\u03c0(-1|x - 1| + 3) - 2) + 1) - 1)^2 + 5(-2(1sin(2\u03c0(-1|x - 1| + 3) - 2) + 1) - 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2sin(3\u03c0(-2(-1x^2 + 5x + 2) + 3) + 1) + 1) + 3)/(2(-2sin(3\u03c0(-2(-1x^2 + 5x + 2) + 3) + 1) + 1) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-1(-2cos(2\u03c0(3|x| - 3) - 3)) + 2))/(2(-1(-2cos(2\u03c0(3|x| - 3) - 3)) + 2) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3cos(3\u03c0(1|((2x - 1)/(-3x - 3))| + 1) - 2) - 3) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2(2sin(2\u03c0((-3x - 1)/(3x - 2)) - 2) + 2) + 3)^2 + 3(2(2sin(2\u03c0((-3x - 1)/(3x - 2)) - 2) + 2) + 3) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(3sin(3\u03c0(1(-5x - 3) + 5) + 3) - 2) - 2)/(3(3sin(3\u03c0(1(-5x - 3) + 5) + 3) - 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-2cos(2\u03c0(-2|(1x - 2) + 3| + 2) - 1) - 1) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(5(2|(-5x^2 + 3x + 3) - 2| + 2) - 2) - 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0((2(2((1x + 3)/(3x - 2)) + 1) + 1)/(-2(2((1x + 3)/(3x - 2)) + 1) + 3))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3sin(2\u03c0(1|(3x^2 + 2x - 5) + 2| + 3) - 2) + 3) + 3)/(2(-3sin(2\u03c0(1|(3x^2 + 2x - 5) + 2| + 3) - 2) + 3) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1sin(2\u03c0(1(-3x^2 - 5x - 1)^2 - 2) + 3)) - 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-2(5((-1x - 2)/(2x - 3))^2 - 5((-1x - 2)/(2x - 3)) - 5)^2 - 2(5((-1x - 2)/(2x - 3))^2 - 5((-1x - 2)/(2x - 3)) - 5) - 5) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3cos(3\u03c0((1(-2|x + 2| + 3) - 1)/(-3(-2|x + 2| + 3) - 2)) + 3)) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1sin(1\u03c0((2(-3x - 5) + 2)/(1(-3x - 5))) - 2) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3(-1cos(1\u03c0((1x - 2)/(2x - 1)) - 3) - 3)^2 - 1(-1cos(1\u03c0((1x - 2)/(2x - 1)) - 3) - 3) + 4) + 3)/(1(-3(-1cos(1\u03c0((1x - 2)/(2x - 1)) - 3) - 3)^2 - 1(-1cos(1\u03c0((1x - 2)/(2x - 1)) - 3) - 3) + 4)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2|(2|(2cos(2\u03c0x - 3) - 3) - 1| + 3) - 2| - 2) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(3sin(2\u03c0(3(2x - 1)^2 - 1(2x - 1) + 1)) - 2)^2 + 3(3sin(2\u03c0(3(2x - 1)^2 - 1(2x - 1) + 1)) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(3|((-3(3sin(3\u03c0x - 1) + 3) - 1)/(2(3sin(3\u03c0x - 1) + 3) - 1)) - 1| + 1))/(3(3|((-3(3sin(3\u03c0x - 1) + 3) - 1)/(2(3sin(3\u03c0x - 1) + 3) - 1)) - 1| + 1) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(5((-3(1sin(3\u03c0x) + 2) + 3)/(-1(1sin(3\u03c0x) + 2) + 3)) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-3cos(2\u03c0(-1(2|x + 1|) + 5) - 1) + 2) - 2)/(-3(-3cos(2\u03c0(-1(2|x + 1|) + 5) - 1) + 2)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(1|(2sin(3\u03c0(2|x + 1| + 2) + 1)) - 2| + 2)^2 + 1(1|(2sin(3\u03c0(2|x + 1| + 2) + 1)) - 2| + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1sin(1\u03c0(-3|(5x^2 + 5x + 4) + 3| + 3) - 2) + 3) + 2| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2sin(1\u03c0((-2(-1x - 3) + 1)/(-1(-1x - 3) + 3)) + 2) + 3)^2 - 3(-2sin(1\u03c0((-2(-1x - 3) + 1)/(-1(-1x - 3) + 3)) + 2) + 3) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-4(-2|(1sin(2\u03c0x + 1) + 1) - 3| + 2)^2 + 5(-2|(1sin(2\u03c0x + 1) + 1) - 3| + 2) + 5) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-3(5(-2cos(2\u03c0x) - 1)^2 - 1(-2cos(2\u03c0x) - 1)) + 5) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0((2((1((3x - 2)/(1x - 3)) + 1)/(1((3x - 2)/(1x - 3)))) - 3)/(1((1((3x - 2)/(1x - 3)) + 1)/(1((3x - 2)/(1x - 3)))) + 3)) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(-3|((3((-1x - 2)/(3x - 2)) - 1)/(-1((-1x - 2)/(3x - 2)) - 1)) + 3|) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(-2|(-1(3|x - 1|)^2 + 1(3|x - 1|) - 4)| + 1) + 3) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-2|(-1(3cos(2\u03c0x + 1) - 3)^2 - 1(3cos(2\u03c0x + 1) - 3) + 2) + 3| - 1) - 1)/(-3(-2|(-1(3cos(2\u03c0x + 1) - 3)^2 - 1(3cos(2\u03c0x + 1) - 3) + 2) + 3| - 1) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3\u03c0(-1|(-3((2x - 1)/(3x + 1)) + 4) + 2| - 2) + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-2|(-2sin(1\u03c0(3|x - 2|) - 3) - 1) - 3| - 2) - 2| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-1cos(1\u03c0(-1|(1x - 5) - 3| - 3) + 2) + 2) - 3)/(3(-1cos(1\u03c0(-1|(1x - 5) - 3| - 3) + 2) + 2) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-4(2cos(1\u03c0(5x^2 + 2x + 2) + 3) - 2)^2 + 2(2cos(1\u03c0(5x^2 + 2x + 2) + 3) - 2) - 5) - 2)/(-3(-4(2cos(1\u03c0(5x^2 + 2x + 2) + 3) - 2)^2 + 2(2cos(1\u03c0(5x^2 + 2x + 2) + 3) - 2) - 5) + 4) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((-1(1|(1cos(1\u03c0x) + 2) - 3| + 1) + 3)/(-3(1|(1cos(1\u03c0x) + 2) - 3| + 1) + 2)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(2|(4(-1sin(2\u03c0x + 1) - 3) - 3)| - 1)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2(-4(-1cos(2\u03c0x + 1) + 1)^2 - 2)^2 - 4(-4(-1cos(2\u03c0x + 1) + 1)^2 - 2) - 1)^2 - 2(2(-4(-1cos(2\u03c0x + 1) + 1)^2 - 2)^2 - 4(-4(-1cos(2\u03c0x + 1) + 1)^2 - 2) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2sin(2\u03c0(-2|(1x^2 - 5) - 1| + 1) + 1) - 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3\u03c0(1|((-2(-1|x| + 2))/(1(-1|x| + 2) + 3))| + 3) + 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-1(-2|(3x - 1) + 1| + 3)^2 - 2(-2|(3x - 1) + 1| + 3) + 2)| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2((-1(-3|(-3cos(1\u03c0x - 2) + 2) - 2| + 3) + 2)/(1(-3|(-3cos(1\u03c0x - 2) + 2) - 2| + 3) + 2))^2 + 1((-1(-3|(-3cos(1\u03c0x - 2) + 2) - 2| + 3) + 2)/(1(-3|(-3cos(1\u03c0x - 2) + 2) - 2| + 3) + 2)) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-1sin(1\u03c0(-2|(-4x^2 - 3x - 5) - 1|) + 1) + 2) - 1| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3|(-3|(1sin(2\u03c0x - 3) + 2) - 2| - 2)| + 3) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((1(-1(-1sin(1\u03c0x - 3) + 2) + 3) + 2)/(-3(-1(-1sin(1\u03c0x - 3) + 2) + 3) - 2)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(-2|(3(2x + 3) + 3) + 3| - 3) - 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2((-2(1(3cos(2\u03c0x) - 1)^2 + 5(3cos(2\u03c0x) - 1) + 4) - 1)/(-1(1(3cos(2\u03c0x) - 1)^2 + 5(3cos(2\u03c0x) - 1) + 4) + 2)) - 2)/(1((-2(1(3cos(2\u03c0x) - 1)^2 + 5(3cos(2\u03c0x) - 1) + 4) - 1)/(-1(1(3cos(2\u03c0x) - 1)^2 + 5(3cos(2\u03c0x) - 1) + 4) + 2)) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1|(-1(1x^2 - 1x + 1)^2 + 4(1x^2 - 1x + 1) - 2) - 1| - 2) - 3| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(1|(-2sin(1\u03c0(-1x + 2) + 3)) - 1| + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((-1(2sin(2\u03c0(1x^2 - 2x - 3) - 2) + 3) - 3)/(2(2sin(2\u03c0(1x^2 - 2x - 3) - 2) + 3) - 1)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(1\u03c0(1|(5(1|x - 2| - 2)^2 - 3(1|x - 2| - 2) + 1) - 2|) - 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(3cos(1\u03c0(-1((3x + 3)/(1x)) + 2) - 3) + 1) + 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((-2(3(2cos(1\u03c0x + 2) + 1) + 4) - 1)/(-3(3(2cos(1\u03c0x + 2) + 1) + 4) - 1)) - 1| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0(-2|(2(-2x - 5)^2 + 1) + 2| + 3) + 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(4(-1(-3cos(3\u03c0x) - 3) - 1) - 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1|(-2cos(1\u03c0((2x - 1)/(-1x + 3))) + 3) + 3|) + 3)/(3(-1|(-2cos(1\u03c0((2x - 1)/(-1x + 3))) + 3) + 3|) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-1sin(1\u03c0(-4((-1x + 2)/(2x + 2))^2 + 4((-1x + 2)/(2x + 2)) + 4) - 3) - 3) - 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-3(-3|(2sin(3\u03c0x) - 3) + 1|)^2 + 1(-3|(2sin(3\u03c0x) - 3) + 1|) - 2)^2 + 3(-3(-3|(2sin(3\u03c0x) - 3) + 1|)^2 + 1(-3|(2sin(3\u03c0x) - 3) + 1|) - 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3((1(-2cos(3\u03c0(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3\u03c0(-3x - 4) - 2) - 1) + 2)) + 3)/(-2((1(-2cos(3\u03c0(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3\u03c0(-3x - 4) - 2) - 1) + 2)) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(1|(4(3sin(3\u03c0x + 2) - 1)^2 - 5(3sin(3\u03c0x + 2) - 1)) - 1| - 2)^2 - 2(1|(4(3sin(3\u03c0x + 2) - 1)^2 - 5(3sin(3\u03c0x + 2) - 1)) - 1| - 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(5(3sin(2\u03c0(-4x + 3) - 3) - 1)^2 + 1(3sin(2\u03c0(-4x + 3) - 3) - 1) + 4) - 1)/(-1(5(3sin(2\u03c0(-4x + 3) - 3) - 1)^2 + 1(3sin(2\u03c0(-4x + 3) - 3) - 1) + 4) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2((-2(2sin(3\u03c0((2x - 3)/(3x + 2)) + 2) - 1) - 1)/(-2(2sin(3\u03c0((2x - 3)/(3x + 2)) + 2) - 1) - 2)) - 3)/(-1((-2(2sin(3\u03c0((2x - 3)/(3x + 2)) + 2) - 1) - 1)/(-2(2sin(3\u03c0((2x - 3)/(3x + 2)) + 2) - 1) - 2)) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2(1(-2sin(2\u03c0x)) - 4) + 4) - 3| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(-1(-3|(-2|x + 3| - 1)| - 3) + 5)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1sin(1\u03c0(1(-4x - 1) + 2) + 2) + 1)^2 - 3(1sin(1\u03c0(1(-4x - 1) + 2) + 2) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1\u03c0(-3(-1(2x^2 - 1x - 2)^2 + 4(2x^2 - 1x - 2) - 5) + 5) + 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-3sin(3\u03c0(-2|(-2x - 2) + 3|)) - 2) + 2)/(-2(-3sin(3\u03c0(-2|(-2x - 2) + 3|)) - 2) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(4((1(1sin(2\u03c0x + 3) - 1))/(3(1sin(2\u03c0x + 3) - 1) - 2))^2 + 4((1(1sin(2\u03c0x + 3) - 1))/(3(1sin(2\u03c0x + 3) - 1) - 2)) + 3)| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(-1(-1|(2x - 5) - 2| - 2)^2 + 2) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(-2(-2(3|x + 2| + 2) + 5)^2 - 1(-2(3|x + 2| + 2) + 5) + 5) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(5(-4(-1sin(3\u03c0x - 3))^2 - 1)^2 - 1)^2 + 5(5(-4(-1sin(3\u03c0x - 3))^2 - 1)^2 - 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-3|(-1sin(1\u03c0((-3x)/(1x + 1)) - 3) + 3) + 1| - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(2|(-2cos(3\u03c0((-2x + 3)/(-2x - 2)) - 2) - 2) + 3| + 1) - 2)/(-3(2|(-2cos(3\u03c0((-2x + 3)/(-2x - 2)) - 2) - 2) + 3| + 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0(4(3(5x^2 - 4x + 5)^2 - 1(5x^2 - 4x + 5) + 4) - 3) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1sin(1\u03c0(-2|((3x - 1)/(3x + 1)) + 1| + 2) - 3) + 1) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(-1|(3(-2|x + 3| + 3)^2 - 4(-2|x + 3| + 3) - 4) - 2| - 2)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(5((-3(-2cos(1\u03c0x + 3) + 3) - 3)/(3(-2cos(1\u03c0x + 3) + 3) - 2)) + 5) + 1| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(3|(-1|(3|x - 1| + 2) - 1|) + 1| + 1) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-2|((-2(-3cos(2\u03c0x + 1) - 2) - 3)/(3(-3cos(2\u03c0x + 1) - 2)))| + 3)^2 - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3|(2cos(1\u03c0(2x^2 - 5x + 3) + 1)) - 1|) - 3| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-1(2|(1sin(2\u03c0x) + 1) - 3| - 2))/(-1(2|(1sin(2\u03c0x) + 1) - 3| - 2) - 1)) + 2| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(1((3(3sin(1\u03c0x - 2) + 1) + 2)/(2(3sin(1\u03c0x - 2) + 1) + 3))^2 + 5((3(3sin(1\u03c0x - 2) + 1) + 2)/(2(3sin(1\u03c0x - 2) + 1) + 3)) - 4) + 2)/(-2(1((3(3sin(1\u03c0x - 2) + 1) + 2)/(2(3sin(1\u03c0x - 2) + 1) + 3))^2 + 5((3(3sin(1\u03c0x - 2) + 1) + 2)/(2(3sin(1\u03c0x - 2) + 1) + 3)) - 4) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3(3cos(3\u03c0((-1x + 3)/(2x)) - 3)) + 3) - 2)/(2(-3(3cos(3\u03c0((-1x + 3)/(2x)) - 3)) + 3)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-2cos(2\u03c0(-3|(-4x - 5) - 1| + 1) - 2) + 2) - 2| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-3sin(3\u03c0((2(-2x^2 - 4x - 3) + 3)/(1(-2x^2 - 4x - 3) + 3))) + 2)^2 - 4(-3sin(3\u03c0((2(-2x^2 - 4x - 3) + 3)/(1(-2x^2 - 4x - 3) + 3))) + 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-4(-4(-2sin(3\u03c0x + 2) - 2)^2 - 1(-2sin(3\u03c0x + 2) - 2) + 4) - 2))/(1(-4(-4(-2sin(3\u03c0x + 2) - 2)^2 - 1(-2sin(3\u03c0x + 2) - 2) + 4) - 2) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3\u03c0(4(-5((-3x + 3)/(1x + 2)) + 4) + 1) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((-3(1sin(1\u03c0(-3x + 5) + 1) - 2) - 2)/(-1(1sin(1\u03c0(-3x + 5) + 1) - 2) - 2))| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1((-1(-3sin(1\u03c0x - 1) - 2) - 3)/(1(-3sin(1\u03c0x - 1) - 2) - 3)) + 1) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3(3sin(3\u03c0(-2|x - 3| + 2) - 1) - 2) + 3)/(1(3sin(3\u03c0(-2|x - 3| + 2) - 1) - 2) + 1))^2 - 2((-3(3sin(3\u03c0(-2|x - 3| + 2) - 1) - 2) + 3)/(1(3sin(3\u03c0(-2|x - 3| + 2) - 1) - 2) + 1)) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(4(1cos(3\u03c0((-2x - 3)/(-1x + 3)))) - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(2\u03c0((1(1((2x + 2)/(-2x + 3))) + 1)/(-3(1((2x + 2)/(-2x + 3))) + 2)) - 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-3cos(3\u03c0(-1((3x - 2)/(3x + 1))^2 - 4((3x - 2)/(3x + 1)) - 4) + 2)) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-2e^(-1(3|(-2x^2 + 3x + 1)| - 2))) - 3)/(-2(-2e^(-1(3|(-2x^2 + 3x + 1)| - 2)))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(4(3sin(1\u03c0(-5x^2 + 1x + 3) - 3) - 1) - 3) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(4(2sin(1\u03c0(-1|x + 3|) + 1) + 3)^2 + 5(2sin(1\u03c0(-1|x + 3|) + 1) + 3) - 1) + 2)/(2(4(2sin(1\u03c0(-1|x + 3|) + 1) + 3)^2 + 5(2sin(1\u03c0(-1|x + 3|) + 1) + 3) - 1) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2sin(2\u03c0(2|((-3x + 2)/(2x - 1)) - 1|) - 3) - 2)^2 + 3(-2sin(2\u03c0(2|((-3x + 2)/(2x - 1)) - 1|) - 3) - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0(-3(3(-1x^2 + 1x - 4) + 2) - 2) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(2cos(3\u03c0((-3((-2x)/(2x + 3)) + 1)/(2((-2x)/(2x + 3)) - 1)) - 1) - 1) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(5((2(-2cos(2\u03c0x - 3) + 3) - 2)/(-3(-2cos(2\u03c0x - 3) + 3) - 2))^2 + 1((2(-2cos(2\u03c0x - 3) + 3) - 2)/(-3(-2cos(2\u03c0x - 3) + 3) - 2))) + 2| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(-2|(2((1x - 1)/(1x + 2))^2 - 2((1x - 1)/(1x + 2)) + 4) - 1| - 1)) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(4(-3(-2cos(3\u03c0x + 1)) - 5)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(3(1((-3x - 3)/(1x + 2))^2 + 5((-3x - 3)/(1x + 2)) + 2) + 4) - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1|(5(3sin(1\u03c0x) + 3) - 5) - 1| + 2)^2 - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|(-1(1|x + 2| - 2)) + 2| - 2)^2 - 1(1|(-1(1|x + 2| - 2)) + 2| - 2) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(4(2cos(3\u03c0(-3x - 5)))^2 + 4(2cos(3\u03c0(-3x - 5))) + 3)^2 - 2(4(2cos(3\u03c0(-3x - 5)))^2 + 4(2cos(3\u03c0(-3x - 5))) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-2|(1cos(2\u03c0((3x - 1)/(-2x - 1)) + 1) - 2) - 3| - 2) + 2)/(-1(-2|(1cos(2\u03c0((3x - 1)/(-2x - 1)) + 1) - 2) - 3| - 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(1|(-1(5x^2 + 4x - 3) + 3) - 2| - 3)^2 - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1cos(3\u03c0((2(1|x - 1|) - 3)/(1(1|x - 1|))) - 3) - 2)|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-1(-2(-3cos(1\u03c0x) + 1)^2 + 2(-3cos(1\u03c0x) + 1) - 4)^2 + 5) + 3)/(-2(-1(-2(-3cos(1\u03c0x) + 1)^2 + 2(-3cos(1\u03c0x) + 1) - 4)^2 + 5) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0((-3(-1|(-1x + 2) - 2| - 2) - 2)/(-2(-1|(-1x + 2) - 2| - 2) - 3)) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(2|(-1|(3cos(1\u03c0x - 3) + 2) + 1|) - 1| + 1) + 3)/(-2(2|(-1|(3cos(1\u03c0x - 3) + 2) + 1|) - 1| + 1) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3(-1sin(2\u03c0(-3x^2 - 1x - 5) + 2) + 1)^2 + 4(-1sin(2\u03c0(-3x^2 - 1x - 5) + 2) + 1) + 4) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-2cos(1\u03c0(-4((1x + 3)/(-1x - 2))^2 + 3((1x + 3)/(-1x - 2)) + 5) + 2) - 3) - 2| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-4((-1(3sin(2\u03c0x + 2) - 3) - 2)/(-2(3sin(2\u03c0x + 2) - 3) - 1)) - 5)^2 + 3(-4((-1(3sin(2\u03c0x + 2) - 3) - 2)/(-2(3sin(2\u03c0x + 2) - 3) - 1)) - 5) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(3sin(3\u03c0(4((2x - 2)/(2x + 1)) + 5) + 2)) + 1| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(2cos(2\u03c0(5(-4x^2 + 2x - 1) + 5)) - 2) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1cos(3\u03c0((2((-2x + 3)/(2x + 1)) + 1)/(3((-2x + 3)/(2x + 1)) - 1)) - 2) - 2) - 2| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(3\u03c0(5(-4((-2x + 2)/(2x - 1)) - 3)^2 + 1(-4((-2x + 2)/(2x - 1)) - 3) + 2) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-1|(1|(-2x^2 - 3x + 1) + 3| + 1) - 2| + 1))/(1(-1|(1|(-2x^2 - 3x + 1) + 3| + 1) - 2| + 1) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-2((-2(2|x| - 1) + 2)/(-1(2|x| - 1) - 1))^2 - 5((-2(2|x| - 1) + 2)/(-1(2|x| - 1) - 1)) - 5) - 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2sin(2\u03c0(-5(1x)^2 - 1(1x) - 1) - 3)) - 1| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(-1|(-3|(-1|x - 2|) - 2| + 2)| - 3) + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-5((2(-3sin(3\u03c0x + 2) - 1) - 3)/(-3(-3sin(3\u03c0x + 2) - 1) - 1))^2 - 5) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0((-2((1((-2x - 3)/(2x + 2)) - 1)/(2((-2x - 3)/(2x + 2)) + 2)) - 2)/(-2((1((-2x - 3)/(2x + 2)) - 1)/(2((-2x - 3)/(2x + 2)) + 2)) + 3))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(3|(-4(-1x^2 - 5x - 2)^2 - 3(-1x^2 - 5x - 2) + 5)|) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((-1(-2(1sin(3\u03c0x - 1)) + 5) - 2)/(3(-2(1sin(3\u03c0x - 1)) + 5) - 2)) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((3(3cos(1\u03c0(4x + 3) - 3)) + 1)/(-1(3cos(1\u03c0(4x + 3) - 3)) + 2)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((2(4(-2sin(3\u03c0x) + 3)^2 + 4(-2sin(3\u03c0x) + 3) - 5) + 3)/(-2(4(-2sin(3\u03c0x) + 3)^2 + 4(-2sin(3\u03c0x) + 3) - 5) + 3))^2 - 3((2(4(-2sin(3\u03c0x) + 3)^2 + 4(-2sin(3\u03c0x) + 3) - 5) + 3)/(-2(4(-2sin(3\u03c0x) + 3)^2 + 4(-2sin(3\u03c0x) + 3) - 5) + 3)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-5(-1|(-5x^2 - 4x + 1)| + 1)^2) - 2| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-1sin(2\u03c0(4((2x)/(2x + 3))^2 - 2((2x)/(2x + 3)) - 2) - 1) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2((-2(2cos(3\u03c0x - 3) - 1) - 3)/(-3(2cos(3\u03c0x - 3) - 1) + 4)) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1(2cos(1\u03c0(-2x + 3) + 2) - 1)) - 1| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(5(1|((-1x - 2)/(-3x + 2)) - 1| - 1)^2 - 1(1|((-1x - 2)/(-3x + 2)) - 1| - 1) + 2)^2 + 4(5(1|((-1x - 2)/(-3x + 2)) - 1| - 1)^2 - 1(1|((-1x - 2)/(-3x + 2)) - 1| - 1) + 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1cos(2\u03c0(2|(3x + 2) - 2| - 3) - 2)) - 1| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(1(-2|(-3|x + 3| + 3) + 3|)) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(-5(-2|(-2x - 1) - 3| + 1)^2 - 4(-2|(-2x - 1) - 3| + 1) - 2)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2sin(2\u03c0(-3|((-2x + 1)/(3x)) + 2| + 3) + 3) + 2)^2 + 4(2sin(2\u03c0(-3|((-2x + 1)/(3x)) + 2| + 3) + 3) + 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(1|(-1sin(1\u03c0((-2x)/(1x + 2)) - 1) + 1) + 3| + 3) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1((3(3(1cos(2\u03c0x + 3) + 3)^2 + 3(1cos(2\u03c0x + 3) + 3) - 5) - 3)/(-2(3(1cos(2\u03c0x + 3) + 3)^2 + 3(1cos(2\u03c0x + 3) + 3) - 5))) + 3)/(2((3(3(1cos(2\u03c0x + 3) + 3)^2 + 3(1cos(2\u03c0x + 3) + 3) - 5) - 3)/(-2(3(1cos(2\u03c0x + 3) + 3)^2 + 3(1cos(2\u03c0x + 3) + 3) - 5)))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((-3(1cos(1\u03c0(-1x + 5)) - 2) - 2)/(-2(1cos(1\u03c0(-1x + 5)) - 2) - 3))^2 - 1((-3(1cos(1\u03c0(-1x + 5)) - 2) - 2)/(-2(1cos(1\u03c0(-1x + 5)) - 2) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0((1((1(-1x + 2) + 3)/(1(-1x + 2) - 3)) + 2)/(-3((1(-1x + 2) + 3)/(1(-1x + 2) - 3)) - 1))) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(5(4(3cos(2\u03c0x - 3) + 2) - 5)^2 - 5(4(3cos(2\u03c0x - 3) + 2) - 5) + 5) - 1)/(1(5(4(3cos(2\u03c0x - 3) + 2) - 5)^2 - 5(4(3cos(2\u03c0x - 3) + 2) - 5) + 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3((-3(-1(1sin(2\u03c0x - 2) - 3)^2 + 4(1sin(2\u03c0x - 2) - 3) - 4) + 2)/(1(-1(1sin(2\u03c0x - 2) - 3)^2 + 4(1sin(2\u03c0x - 2) - 3) - 4))))/(3((-3(-1(1sin(2\u03c0x - 2) - 3)^2 + 4(1sin(2\u03c0x - 2) - 3) - 4) + 2)/(1(-1(1sin(2\u03c0x - 2) - 3)^2 + 4(1sin(2\u03c0x - 2) - 3) - 4))) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(3sin(2\u03c0(-3(3|x + 1| - 2) + 3) + 1) + 2)^2 - 2(3sin(2\u03c0(-3(3|x + 1| - 2) + 3) + 1) + 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(5((-1(5x - 4) - 3)/(-1(5x - 4) + 2))^2 - 4((-1(5x - 4) - 3)/(-1(5x - 4) + 2)) + 1) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4((1(-1|(1sin(3\u03c0x + 2) - 3) + 3| + 2))/(-1(-1|(1sin(3\u03c0x + 2) - 3) + 3| + 2) + 2))^2 - 3((1(-1|(1sin(3\u03c0x + 2) - 3) + 3| + 2))/(-1(-1|(1sin(3\u03c0x + 2) - 3) + 3| + 2) + 2)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(3|(-1(2sin(3\u03c0x - 2) - 3) + 2)| + 2)^2 + 5(3|(-1(2sin(3\u03c0x - 2) - 3) + 2)| + 2) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1(1(2cos(2\u03c0x + 1) + 1)^2 - 4(2cos(2\u03c0x + 1) + 1) + 1)^2 + 1(1(2cos(2\u03c0x + 1) + 1)^2 - 4(2cos(2\u03c0x + 1) + 1) + 1) + 5) - 3)/(1(-1(1(2cos(2\u03c0x + 1) + 1)^2 - 4(2cos(2\u03c0x + 1) + 1) + 1)^2 + 1(1(2cos(2\u03c0x + 1) + 1)^2 - 4(2cos(2\u03c0x + 1) + 1) + 1) + 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2sin(2\u03c0(-4(1x - 5) + 1) - 1) + 1)^2 + 5(2sin(2\u03c0(-4(1x - 5) + 1) - 1) + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-3sin(1\u03c0((1((3x + 2)/(-3x)) - 3)/(-3((3x + 2)/(-3x)))) - 2) + 2) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3((-1(-2cos(1\u03c0(1x^2 + 5x + 3) - 1) + 3))/(-2(-2cos(1\u03c0(1x^2 + 5x + 3) - 1) + 3) - 3)) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3sin(1\u03c0(1|(2|x| + 3)| + 2) - 2) + 2) - 1)/(3(3sin(1\u03c0(1|(2|x| + 3)| + 2) - 2) + 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5((2(1(3sin(1\u03c0x) - 3)^2 - 5(3sin(1\u03c0x) - 3) + 5) - 2)/(1(1(3sin(1\u03c0x) - 3)^2 - 5(3sin(1\u03c0x) - 3) + 5) - 3))^2 + 5((2(1(3sin(1\u03c0x) - 3)^2 - 5(3sin(1\u03c0x) - 3) + 5) - 2)/(1(1(3sin(1\u03c0x) - 3)^2 - 5(3sin(1\u03c0x) - 3) + 5) - 3)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-1|(-4(-3cos(1\u03c0x + 3) + 2) + 2) - 1| + 2))/(-1(-1|(-4(-3cos(1\u03c0x + 3) + 2) + 2) - 1| + 2) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-4(-1sin(2\u03c0(-4x - 5) - 2))) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-1cos(3\u03c0((1(3x^2 - 4) + 3)/(-2(3x^2 - 4) - 3)) - 1) - 3)^2 - 3(-1cos(3\u03c0((1(3x^2 - 4) + 3)/(-2(3x^2 - 4) - 3)) - 1) - 3) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1\u03c0(-3|((-3((3x)/(1x - 3)) + 2)/(1((3x)/(1x - 3)) - 2)) + 1| + 3) + 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1sin(2\u03c0(-3(-1x^2 + 1x + 5) - 5) - 2) - 3) - 3)/(-1(1sin(2\u03c0(-3(-1x^2 + 1x + 5) - 5) - 2) - 3)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((1(-2(3cos(1\u03c0x - 3) - 2) + 4) - 2)/(2(-2(3cos(1\u03c0x - 3) - 2) + 4) - 3)) + 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(2|((-3(1x^2 + 5x - 2) - 1)/(1(1x^2 + 5x - 2) + 1)) + 3| - 2) - 1| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1(3|(1|x - 1| - 1) - 2|) + 4)^2 - 1(-1(3|(1|x - 1| - 1) - 2|) + 4) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(4(1(-3x^2 + 3x - 1) - 2) + 3) + 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(5(-3cos(3\u03c0(3|x - 2| + 2) - 3) + 3)^2 + 2(-3cos(3\u03c0(3|x - 2| + 2) - 3) + 3) + 5) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|((-2(2|((2x + 2)/(1x + 2)) - 1| + 1) - 3)/(1(2|((2x + 2)/(1x + 2)) - 1| + 1) - 2)) + 3| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(1|(3sin(2\u03c0(-3x - 5) - 3) - 1) - 2|) + 3| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((3(1cos(3\u03c0(-3|x + 3| - 3) - 3) + 1) + 3)/(-2(1cos(3\u03c0(-3|x + 3| - 3) - 3) + 1) + 1)) - 2| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2|(4(-1sin(2\u03c0x - 3) + 1)^2 + 5(-1sin(2\u03c0x - 3) + 1) + 5) + 3| - 2) + 3)/(3(-2|(4(-1sin(2\u03c0x - 3) + 1)^2 + 5(-1sin(2\u03c0x - 3) + 1) + 5) + 3| - 2) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2(-2sin(2\u03c0(-2|x + 1| + 2) + 3) + 1)^2 - 3(-2sin(2\u03c0(-2|x + 1| + 2) + 3) + 1) - 2)^2 - 3(-2(-2sin(2\u03c0(-2|x + 1| + 2) + 3) + 1)^2 - 3(-2sin(2\u03c0(-2|x + 1| + 2) + 3) + 1) - 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(2\u03c0((2(-1|(-1x^2 + 2x - 2) - 3| - 1) - 2)/(1(-1|(-1x^2 + 2x - 2) - 3| - 1) + 3)) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(4(-1|(2sin(2\u03c0x - 1) + 1) + 3| + 3)^2 + 4(-1|(2sin(2\u03c0x - 1) + 1) + 3| + 3) - 2) + 1)/(-1(4(-1|(2sin(2\u03c0x - 1) + 1) + 3| + 3)^2 + 4(-1|(2sin(2\u03c0x - 1) + 1) + 3| + 3) - 2) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-3sin(1\u03c0(5(-2x^2 - 4x + 2) + 4) + 1))^2 + 3(-3sin(1\u03c0(5(-2x^2 - 4x + 2) + 4) + 1)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1((-1((1(2cos(2\u03c0x - 1) + 3) - 1)/(-2(2cos(2\u03c0x - 1) + 3) - 2)) - 1)/(2((1(2cos(2\u03c0x - 1) + 3) - 1)/(-2(2cos(2\u03c0x - 1) + 3) - 2)) - 3))^2 - 1((-1((1(2cos(2\u03c0x - 1) + 3) - 1)/(-2(2cos(2\u03c0x - 1) + 3) - 2)) - 1)/(2((1(2cos(2\u03c0x - 1) + 3) - 1)/(-2(2cos(2\u03c0x - 1) + 3) - 2)) - 3)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3\u03c0(-1|(-1|(-3|x - 1| - 2) + 2| + 2) + 1| - 1) + 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2(1sin(2\u03c0(1|x - 1| + 3) + 2) + 3)^2 - 4)^2 + 1(2(1sin(2\u03c0(1|x - 1| + 3) + 2) + 3)^2 - 4) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(5(-1|(-2|x| - 3) - 1|) - 1) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((3((2(-2sin(1\u03c0x - 2) + 1) + 2)/(-3(-2sin(1\u03c0x - 2) + 1))) - 2)/(-2((2(-2sin(1\u03c0x - 2) + 1) + 2)/(-3(-2sin(1\u03c0x - 2) + 1))) + 3)) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(3|(1(2|x - 1| - 3)^2 - 5(2|x - 1| - 3) - 2) + 2| - 1)^2 + 3(3|(1(2|x - 1| - 3)^2 - 5(2|x - 1| - 3) - 2) + 2| - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(-3(2|(2|x - 2|) - 3| + 2)) + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-5(1cos(3\u03c0(-1x^2 - 2x + 4) + 3) - 2)^2 - 4(1cos(3\u03c0(-1x^2 - 2x + 4) + 3) - 2) - 4) - 2| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-1sin(2\u03c0(-5(-1|x + 2| - 2)^2 - 3(-1|x + 2| - 2) - 2)) + 2) + 3| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-1|(3sin(2\u03c0(1|x|) + 1)) + 1| + 2) + 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-5(-5(1cos(2\u03c0x + 2) - 2)^2 + 5(1cos(2\u03c0x + 2) - 2)) + 5) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(5(-1(1x^2 - 1x - 2)^2 + 5(1x^2 - 1x - 2) + 2)^2 - 4(-1(1x^2 - 1x - 2)^2 + 5(1x^2 - 1x - 2) + 2) + 5) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1|(3(3cos(3\u03c0x + 3) - 2) + 5) - 1| - 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-2|(-1(-1cos(3\u03c0x - 2) - 3)^2 + 3(-1cos(3\u03c0x - 2) - 3) - 3)| + 1) + 1)/(3(-2|(-1(-1cos(3\u03c0x - 2) - 3)^2 + 3(-1cos(3\u03c0x - 2) - 3) - 3)| + 1) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1\u03c0(-2(-5((-3x + 1)/(1x - 3))^2 - 3((-3x + 1)/(1x - 3)) + 4) + 5) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(-1|(-3|(3x)|) + 1| - 3) - 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2cos(2\u03c0(-1(-2x^2 - 4x - 2) - 5)) + 2)^2 - 3(-2cos(2\u03c0(-1(-2x^2 - 4x - 2) - 5)) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(5((3(-1|x - 1| - 3) + 2)/(-2(-1|x - 1| - 3) - 3))^2 + 1) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(4((-1(-1sin(1\u03c0x + 2) + 1) + 2)/(-2(-1sin(1\u03c0x + 2) + 1)))^2 - 1((-1(-1sin(1\u03c0x + 2) + 1) + 2)/(-2(-1sin(1\u03c0x + 2) + 1)))) - 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1cos(1\u03c0(-2(-3x + 2) + 1)) - 3)| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2sin(1\u03c0((2(-5x^2 - 4x + 3) + 3)/(-1(-5x^2 - 4x + 3) + 3)) - 2) - 2)^2 + 4(2sin(1\u03c0((2(-5x^2 - 4x + 3) + 3)/(-1(-5x^2 - 4x + 3) + 3)) - 2) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(1sin(1\u03c0((-3(-1x^2 + 4x) - 1)/(-3(-1x^2 + 4x) - 3)) - 2)) - 1)/(1(1sin(1\u03c0((-3(-1x^2 + 4x) - 1)/(-3(-1x^2 + 4x) - 3)) - 2)) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-2|(1(3cos(1\u03c0x - 3) + 1) - 1) + 3| - 1)^2 - 3(-2|(1(3cos(1\u03c0x - 3) + 1) - 1) + 3| - 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2sin(3\u03c0(3(2x - 3) - 1) - 1)) + 2)/(2(2sin(3\u03c0(3(2x - 3) - 1) - 1)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(1(-5(3x - 3) + 1) + 5) + 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(-2(4(-3x^2 - 2x + 1) - 5) + 1) + 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-3sin(2\u03c0(-3|(4x + 2) + 3| - 1) - 3) - 1) - 1)/(-1(-3sin(2\u03c0(-3|(4x + 2) + 3| - 1) - 3) - 1) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(4(-1|(-3cos(3\u03c0x + 3) + 2) + 2| + 1)^2 - 4(-1|(-3cos(3\u03c0x + 3) + 2) + 2| + 1) - 3)^2 + 5(4(-1|(-3cos(3\u03c0x + 3) + 2) + 2| + 1)^2 - 4(-1|(-3cos(3\u03c0x + 3) + 2) + 2| + 1) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(-3|(-3(2x)^2 - 1(2x) - 3) + 1| + 1) - 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(2cos(1\u03c0(1(-4x - 2) + 2) - 1)) - 3)/(-3(2cos(1\u03c0(1(-4x - 2) + 2) - 1)) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1cos(3\u03c0((1(2x - 3) - 3)/(3(2x - 3) + 2)) + 2) - 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2(1sin(2\u03c0(1x^2 + 3x - 3) - 2) - 1) - 2) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0((3(-3|(4x + 1)| + 2) + 1)/(1(-3|(4x + 1)| + 2) - 3)) + 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2(-2|(2x + 3) + 2| + 3)^2 - 4(-2|(2x + 3) + 2| + 3))^2 + 1(2(-2|(2x + 3) + 2| + 3)^2 - 4(-2|(2x + 3) + 2| + 3)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2|(3cos(1\u03c0(-1|x + 3| + 2) + 1) - 3) - 1| + 3) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(-3((2((-2x + 2)/(2x - 1)))/(3((-2x + 2)/(2x - 1)) - 2))^2 - 1((2((-2x + 2)/(2x - 1)))/(3((-2x + 2)/(2x - 1)) - 2)) - 5) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-3sin(2\u03c0(-1(-4x^2 - 3x - 1)^2 + 2(-4x^2 - 3x - 1) - 2) + 2) + 1) + 1|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(1cos(2\u03c0(1|(1|x| - 1) - 3| + 2)) - 3) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-3cos(2\u03c0(3|((2x - 1)/(-1x - 2)) + 1| - 3) + 2) - 3) + 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1sin(2\u03c0(1((2x - 1)/(-1x - 1)) - 4)) + 1)^2 + 1(1sin(2\u03c0(1((2x - 1)/(-1x - 1)) - 4)) + 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1\u03c0((2((-3(-5x^2 - 5x - 3) - 1)/(3(-5x^2 - 5x - 3) + 2)))/(-2((-3(-5x^2 - 5x - 3) - 1)/(3(-5x^2 - 5x - 3) + 2)) - 2)) + 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(4(1|(2cos(2\u03c0x + 2) - 3) - 1| - 2) + 2)| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-1cos(2\u03c0((1(1x^2 + 3x + 1) - 1)/(-3(1x^2 + 3x + 1))) - 3) - 3)| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(2|((-3(-2x^2 - 4x + 3) - 1)/(-2(-2x^2 - 4x + 3) + 1)) - 2|) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2((2((-3(2sin(3\u03c0x - 1) + 2) - 3)/(-2(2sin(3\u03c0x - 1) + 2) + 1)) + 1)/(1((-3(2sin(3\u03c0x - 1) + 2) - 3)/(-2(2sin(3\u03c0x - 1) + 2) + 1)) + 3))^2 - 3((2((-3(2sin(3\u03c0x - 1) + 2) - 3)/(-2(2sin(3\u03c0x - 1) + 2) + 1)) + 1)/(1((-3(2sin(3\u03c0x - 1) + 2) - 3)/(-2(2sin(3\u03c0x - 1) + 2) + 1)) + 3)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((1(2cos(2\u03c0((-3x + 3)/(-3x + 1)) + 3) - 2) + 1)/(-2(2cos(2\u03c0((-3x + 3)/(-3x + 1)) + 3) - 2) + 3))^2 - 4((1(2cos(2\u03c0((-3x + 3)/(-3x + 1)) + 3) - 2) + 1)/(-2(2cos(2\u03c0((-3x + 3)/(-3x + 1)) + 3) - 2) + 3)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-5(2cos(2\u03c0((-1x + 2)/(-1x - 1)) - 1) - 2)^2 + 4(2cos(2\u03c0((-1x + 2)/(-1x - 1)) - 1) - 2) + 3)^2 + 1(-5(2cos(2\u03c0((-1x + 2)/(-1x - 1)) - 1) - 2)^2 + 4(2cos(2\u03c0((-1x + 2)/(-1x - 1)) - 1) - 2) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1|(-2sin(3\u03c0(2x + 5) - 1) + 1) - 2| - 2)| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0(-5(2((3x + 1)/(-3x - 3)) - 1) + 1) + 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-3|(2cos(1\u03c0((-3x + 2)/(-1x - 2))) + 1) + 1| - 2))/(-1(-3|(2cos(1\u03c0((-3x + 2)/(-1x - 2))) + 1) + 1| - 2) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1(1|(1|x + 1| - 2)| + 2)^2 + 5(1|(1|x + 1| - 2)| + 2) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2cos(3\u03c0((1(4x^2 + 5x + 2) - 2)/(2(4x^2 + 5x + 2) - 3))) + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(3(4(3cos(1\u03c0x - 3) + 2) + 5) - 5) + 2| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2|(-5(-1x^2 + 5x - 1)^2 + 2(-1x^2 + 5x - 1) + 4) + 2| - 3) - 3)/(-2(2|(-5(-1x^2 + 5x - 1)^2 + 2(-1x^2 + 5x - 1) + 4) + 2| - 3) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(1\u03c0((-2(-5x^2 + 3x + 2) - 3)/(-1(-5x^2 + 3x + 2))) + 3) + 3) - 1)/(3(-1sin(1\u03c0((-2(-5x^2 + 3x + 2) - 3)/(-1(-5x^2 + 3x + 2))) + 3) + 3) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(1|(2cos(3\u03c0(-2|x + 2| + 3) - 2) - 2) - 2| + 2)^2 - 2(1|(2cos(3\u03c0(-2|x + 2| + 3) - 2) - 2) - 2| + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1(-1cos(3\u03c0(-3x^2 + 5x - 3) + 2) + 1) - 3) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1|((2(2sin(2\u03c0x - 2) - 2) - 2)/(-1(2sin(2\u03c0x - 2) - 2) - 3)) + 3| + 2)^2 - 4(1|((2(2sin(2\u03c0x - 2) - 2) - 2)/(-1(2sin(2\u03c0x - 2) - 2) - 3)) + 3| + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((-1(2cos(3\u03c0(-1|x + 2| + 2) + 2) + 1) + 3)/(-2(2cos(3\u03c0(-1|x + 2| + 2) + 2) + 1) + 1))^2 + 3((-1(2cos(3\u03c0(-1|x + 2| + 2) + 2) + 1) + 3)/(-2(2cos(3\u03c0(-1|x + 2| + 2) + 2) + 1) + 1)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((-3(-3(3sin(1\u03c0x + 1) - 3)^2 + 3(3sin(1\u03c0x + 1) - 3) - 1) - 3)/(-1(-3(3sin(1\u03c0x + 1) - 3)^2 + 3(3sin(1\u03c0x + 1) - 3) - 1) - 3))^2 + 3((-3(-3(3sin(1\u03c0x + 1) - 3)^2 + 3(3sin(1\u03c0x + 1) - 3) - 1) - 3)/(-1(-3(3sin(1\u03c0x + 1) - 3)^2 + 3(3sin(1\u03c0x + 1) - 3) - 1) - 3)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(3|((-2(2cos(2\u03c0x + 3) + 3) - 3)/(-3(2cos(2\u03c0x + 3) + 3) - 3)) - 3| + 3) - 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-2cos(2\u03c0(1x^2 + 4x + 5) - 2) + 1) + 2)^2 - 4(1(-2cos(2\u03c0(1x^2 + 4x + 5) - 2) + 1) + 2) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1(3(-1sin(1\u03c0x - 2) - 2) - 2) + 3) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(4(-2|(2sin(2\u03c0x - 3) - 3) + 1| - 1)^2 + 1(-2|(2sin(2\u03c0x - 3) - 3) + 1| - 1) - 1) - 3)/(-3(4(-2|(2sin(2\u03c0x - 3) - 3) + 1| - 1)^2 + 1(-2|(2sin(2\u03c0x - 3) - 3) + 1| - 1) - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(3(-2cos(3\u03c0((-2x - 2)/(-1x + 2)) - 1) - 3)) - 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-4((2(3sin(2\u03c0x + 3) + 2) + 3)/(-3(3sin(2\u03c0x + 3) + 2) + 4))^2 + 3((2(3sin(2\u03c0x + 3) + 2) + 3)/(-3(3sin(2\u03c0x + 3) + 2) + 4)) + 1) + 2)/(-1(-4((2(3sin(2\u03c0x + 3) + 2) + 3)/(-3(3sin(2\u03c0x + 3) + 2) + 4))^2 + 3((2(3sin(2\u03c0x + 3) + 2) + 3)/(-3(3sin(2\u03c0x + 3) + 2) + 4)) + 1) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(1(3|(1x + 5) - 1| - 2)^2 - 5(3|(1x + 5) - 1| - 2) + 4) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1|(-5(-3sin(3\u03c0x + 2) - 2) - 1) + 3| + 2) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-1|((1(-1sin(2\u03c0x + 2) - 1) + 1)/(-3(-1sin(2\u03c0x + 2) - 1) - 2)) + 3| - 1) + 1)/(-3(-1|((1(-1sin(2\u03c0x + 2) - 1) + 1)/(-3(-1sin(2\u03c0x + 2) - 1) - 2)) + 3| - 1) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(1(-4(3cos(2\u03c0x - 1) - 1) - 4) + 1) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(2sin(1\u03c0(3(-3|x| - 1) + 2) - 2) + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-4((1(1sin(1\u03c0x) + 2) - 2)/(2(1sin(1\u03c0x) + 2) - 1))) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0((-2((3(-5x + 2) + 3)/(-3(-5x + 2) - 1)) - 1)/(1((3(-5x + 2) + 3)/(-3(-5x + 2) - 1)))) + 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3|(1|(-1cos(2\u03c0x + 2) + 2) - 2| + 3) - 2| + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(5(1|(2|x + 2| - 3) - 1|)^2 + 4(1|(2|x + 2| - 3) - 1|) - 2)^2 + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-4(-1|(1cos(2\u03c0x + 1)) + 3| - 2)^2 - 1(-1|(1cos(2\u03c0x + 1)) + 3| - 2) + 1) + 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2|(3cos(2\u03c0(1x^2 - 3x + 2) + 1)) + 2| - 3)^2 - 1(2|(3cos(2\u03c0(1x^2 - 3x + 2) + 1)) + 2| - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2\u03c0(5(3(-1|x - 3| + 3) - 4)^2 - 5(3(-1|x - 3| + 3) - 4) + 3) + 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-4((2(1cos(3\u03c0x + 3) + 2) - 3)/(-3(1cos(3\u03c0x + 3) + 2)))^2 + 1((2(1cos(3\u03c0x + 3) + 2) - 3)/(-3(1cos(3\u03c0x + 3) + 2))) - 3) + 3| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3(4(1cos(2\u03c0x + 1) - 1) + 5) - 3)^2 + 3(-3(4(1cos(2\u03c0x + 1) - 1) + 5) - 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(-3|(3(-2x^2 - 2x - 4)^2 + 2(-2x^2 - 2x - 4)) + 3| + 1) - 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(1cos(3\u03c0(-3x + 3) + 3)) + 4) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-3sin(1\u03c0(-5(-3x^2 - 3x + 2)) - 2) + 1)|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1cos(2\u03c0(1|(-2x^2 + 2x) + 2| - 3) - 1) - 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((1((3(-2cos(3\u03c0x + 2) + 3) - 3)/(-1(-2cos(3\u03c0x + 2) + 3) - 1)) - 1)/(-3((3(-2cos(3\u03c0x + 2) + 3) - 3)/(-1(-2cos(3\u03c0x + 2) + 3) - 1)) - 1))^2 + 4((1((3(-2cos(3\u03c0x + 2) + 3) - 3)/(-1(-2cos(3\u03c0x + 2) + 3) - 1)) - 1)/(-3((3(-2cos(3\u03c0x + 2) + 3) - 3)/(-1(-2cos(3\u03c0x + 2) + 3) - 1)) - 1)) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-5(2sin(1\u03c0(-3x^2 + 4) + 3) - 2) - 2) - 1)/(-2(-5(2sin(1\u03c0(-3x^2 + 4) + 3) - 2) - 2) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(2|(2cos(3\u03c0x - 3) - 1) + 3| + 2) - 3)^2 + 2(-1(2|(2cos(3\u03c0x - 3) - 1) + 3| + 2) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3|(-1cos(1\u03c0(-3x + 5) - 3) - 1) - 3| - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(5(1(1sin(1\u03c0x + 1) + 1) - 4) - 2))/(-2(5(1(1sin(1\u03c0x + 1) + 1) - 4) - 2) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((3(-3sin(3\u03c0((-2x - 1)/(2x + 2)) + 3)) + 3)/(-3(-3sin(3\u03c0((-2x - 1)/(2x + 2)) + 3)) - 2))^2 + 4((3(-3sin(3\u03c0((-2x - 1)/(2x + 2)) + 3)) + 3)/(-3(-3sin(3\u03c0((-2x - 1)/(2x + 2)) + 3)) - 2)) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1cos(2\u03c0(4((3x)/(1x - 3)) + 5)) - 1) - 1)/(3(-1cos(2\u03c0(4((3x)/(1x - 3)) + 5)) - 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(2cos(3\u03c0(1(5x^2 + 1x - 1) - 5) - 3) + 2) - 3)/(-2(2cos(3\u03c0(1(5x^2 + 1x - 1) - 5) - 3) + 2) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0(2(-2(1|x - 1| + 3) - 3)) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0((-2(4((1x)/(-1x + 2)) + 2) + 3)/(-2(4((1x)/(-1x + 2)) + 2))) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(3\u03c0(1|((2(5x + 5) + 3)/(-2(5x + 5) + 3)) - 2| + 1) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-2(3((-1x)/(-2x + 3)) - 2)^2 + 5) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(2|(1|(2x - 2) + 1| - 2) - 2| - 1) - 1| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2sin(1\u03c0(2|(2|x - 1| - 1) + 1|) - 3) + 3)^2 + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3\u03c0(1(3|(1x + 1) + 3| - 1)^2 - 4(3|(1x + 1) + 3| - 1) + 1) - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-1(1(-1x^2 + 1x + 5) + 4)^2 - 5(1(-1x^2 + 1x + 5) + 4)) - 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2sin(2\u03c0(-2(2x - 3)^2 + 1(2x - 3) - 3)) + 1) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3((2(-2cos(1\u03c0((2x + 2)/(1x + 2))) - 2) - 1)/(3(-2cos(1\u03c0((2x + 2)/(1x + 2))) - 2) - 1)) + 2)/(-3((2(-2cos(1\u03c0((2x + 2)/(1x + 2))) - 2) - 1)/(3(-2cos(1\u03c0((2x + 2)/(1x + 2))) - 2) - 1)) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3sin(3\u03c0(5(-2|x - 1|) - 1)) + 1)^2 + 4(-3sin(3\u03c0(5(-2|x - 1|) - 1)) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(4(2|(1x^2 + 5x + 1) + 3| - 2)^2 - 3(2|(1x^2 + 5x + 1) + 3| - 2) - 5) + 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3|(-1(-1cos(2\u03c0x + 3))^2 - 3(-1cos(2\u03c0x + 3))) + 3| - 2) + 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1|(-1(-3sin(3\u03c0x - 2) + 2) + 2) - 1| + 3)^2 + 2(-1|(-1(-3sin(3\u03c0x - 2) + 2) + 2) - 1| + 3) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0(4(-3|(2|x + 3| - 1) + 3|) + 5) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-2(-2sin(1\u03c0(2|x| - 1) - 1)) - 2) - 2| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-3cos(2\u03c0(3(5x^2 - 3x + 4) - 3) - 2) + 1) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(2(-3|(-2|x - 2| - 1) - 2| - 2) - 2) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-4(-1|(-1|x - 3| + 1) + 3| + 3) + 1)^2 - 4(-4(-1|(-1|x - 3| + 1) + 3| + 3) + 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2cos(2\u03c0(-2|(3x - 4) + 2| + 2) + 2) + 2)^2 - 5(-2cos(2\u03c0(-2|(3x - 4) + 2| + 2) + 2) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(1\u03c0(3|(-3|(-5x + 1) - 1| - 3)| + 3) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(4(-1|(-1sin(3\u03c0x + 1)) - 2|)^2 - 1(-1|(-1sin(3\u03c0x + 1)) - 2|))|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|((-3(1cos(2\u03c0x - 2)) + 2)/(-2(1cos(2\u03c0x - 2)))) - 2| - 2)^2 + 5(1|((-3(1cos(2\u03c0x - 2)) + 2)/(-2(1cos(2\u03c0x - 2)))) - 2| - 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-2|(-1(-4x) - 3) + 1| + 3) + 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-1cos(3\u03c0(-2(-3x^2 - 1) + 3) - 3)) + 2| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1cos(2\u03c0(-1((2x - 3)/(2x + 2)) + 1) + 3) + 1)^2 + 1(-1cos(2\u03c0(-1((2x - 3)/(2x + 2)) + 1) + 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0(-1((2(-4x^2 - 4x - 5) - 3)/(2(-4x^2 - 4x - 5) + 3)) - 3) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-3sin(3\u03c0(1|(-3x + 3) + 2| + 2) + 3) - 3) - 2)/(3(-3sin(3\u03c0(1|(-3x + 3) + 2| + 2) + 3) - 3) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2cos(1\u03c0(-3(5x - 3) + 4) - 3) - 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2|(3(1x^2 - 3x - 1)^2 - 1(1x^2 - 3x - 1) - 2)| + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-2(4(-4x^2 + 2)^2 - 1(-4x^2 + 2) - 3) + 3)|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-1((1(1cos(1\u03c0x + 1) - 2) - 1)/(-1(1cos(1\u03c0x + 1) - 2) + 2))^2 + 4)^2 - 4(-1((1(1cos(1\u03c0x + 1) - 2) - 1)/(-1(1cos(1\u03c0x + 1) - 2) + 2))^2 + 4) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(4(1(-3sin(2\u03c0x + 1) - 3) - 3)^2 - 4))/(1(4(1(-3sin(2\u03c0x + 1) - 3) - 3)^2 - 4) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(4(2cos(1\u03c0(2x^2 - 2x - 2) - 1) - 1) + 2) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-3cos(2\u03c0(5(-1x^2 - 5x - 5)^2 + 5(-1x^2 - 5x - 5) + 1)) + 3)^2 + 2(-3cos(2\u03c0(5(-1x^2 - 5x - 5)^2 + 5(-1x^2 - 5x - 5) + 1)) + 3) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2\u03c0(2|(1(-1x)^2 + 3(-1x) - 5)| + 2) - 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(2\u03c0(5(2|(-2x^2 - 2x + 4) - 2| - 1)^2 - 2(2|(-2x^2 - 2x + 4) - 2| - 1) + 4) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-4(-4(3cos(3\u03c0x + 3) - 2) + 4) - 5) - 3)/(1(-4(-4(3cos(3\u03c0x + 3) - 2) + 4) - 5) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5((1(-1cos(1\u03c0(-4x - 5) + 2) - 2) - 3)/(-1(-1cos(1\u03c0(-4x - 5) + 2) - 2)))^2 - 5((1(-1cos(1\u03c0(-4x - 5) + 2) - 2) - 3)/(-1(-1cos(1\u03c0(-4x - 5) + 2) - 2))) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(5(1cos(2\u03c0(4x^2 + 1x) - 2) + 3) + 2) - 1|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-2|(-2sin(2\u03c0(-1x^2 - 3x) + 3) - 2) - 2|) + 2| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0(-2((2(-5x^2 - 2x + 5) - 1)/(2(-5x^2 - 2x + 5) + 1)) - 3) + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-1cos(2\u03c0(3(1x^2 - 2x)^2 + 5(1x^2 - 2x) + 1) + 1))^2 + 5(-1cos(2\u03c0(3(1x^2 - 2x)^2 + 5(1x^2 - 2x) + 1) + 1)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-5(-2sin(1\u03c0((-2x)/(1x - 2)) - 1) - 3)^2 - 2(-2sin(1\u03c0((-2x)/(1x - 2)) - 1) - 3) + 5) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(4(-2|(2x + 2) - 2|)^2 + 1(-2|(2x + 2) - 2|) - 5) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(3|(-3((1x - 1)/(-1x - 1))^2 + 1((1x - 1)/(-1x - 1)) - 1) + 2| - 2) + 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2\u03c0(4(4(3|x + 2| - 2)) - 1) - 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(3((2(-2|x + 3| - 1) + 2)/(3(-2|x + 3| - 1))) + 3) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((-2(2(-2|x - 1| + 3)^2 - 3(-2|x - 1| + 3) - 5) - 3)/(1(2(-2|x - 1| + 3)^2 - 3(-2|x - 1| + 3) - 5))) + 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-3(-2(2sin(3\u03c0x) - 1) + 2)^2 + 1(-2(2sin(3\u03c0x) - 1) + 2) - 5) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2|(-2|(2sin(3\u03c0x + 2) - 1) + 2| + 3) + 3| + 1))/(-2(-2|(-2|(2sin(3\u03c0x + 2) - 1) + 2| + 3) + 3| + 1) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-5(1sin(3\u03c0(1x^2 - 3x + 1) - 1) - 2) - 4) - 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2cos(2\u03c0(1(-2x^2 + 2x + 2) + 4)) + 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-5(-3|(1cos(2\u03c0x + 3) - 3) + 3| - 3)^2 - 3(-3|(1cos(2\u03c0x + 3) - 3) + 3| - 3) + 3) - 3)/(3(-5(-3|(1cos(2\u03c0x + 3) - 3) + 3| - 3)^2 - 3(-3|(1cos(2\u03c0x + 3) - 3) + 3| - 3) + 3) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3((3(4(-3sin(2\u03c0x + 1) - 1)^2 + 3(-3sin(2\u03c0x + 1) - 1) - 4) + 1)/(-1(4(-3sin(2\u03c0x + 1) - 1)^2 + 3(-3sin(2\u03c0x + 1) - 1) - 4) - 2)))/(1((3(4(-3sin(2\u03c0x + 1) - 1)^2 + 3(-3sin(2\u03c0x + 1) - 1) - 4) + 1)/(-1(4(-3sin(2\u03c0x + 1) - 1)^2 + 3(-3sin(2\u03c0x + 1) - 1) - 4) - 2)) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-5((-2(1cos(3\u03c0x + 1) + 1))/(3(1cos(3\u03c0x + 1) + 1) + 2))^2 + 2((-2(1cos(3\u03c0x + 1) + 1))/(3(1cos(3\u03c0x + 1) + 1) + 2)) + 4) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(-1(-4(-1x - 3) + 2)^2 + 3(-4(-1x - 3) + 2) + 3) + 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(3cos(2\u03c0((-2((3x)/(3x + 3)) + 3)/(1((3x)/(3x + 3)) + 3)) - 2) - 2) - 1| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1(5(-3cos(3\u03c0x + 3) - 3)^2 - 2)^2 + 3(5(-3cos(3\u03c0x + 3) - 3)^2 - 2) + 1) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-2sin(3\u03c0(-4(1x + 3) - 1) + 1) + 3) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((1(-2sin(1\u03c0((2x - 3)/(2x)) - 1) - 3) - 3)/(2(-2sin(1\u03c0((2x - 3)/(2x)) - 1) - 3) - 1)) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(1(3cos(2\u03c0(-3x^2 + 5x - 4) - 1)) + 5) + 2)/(-1(1(3cos(2\u03c0(-3x^2 + 5x - 4) - 1)) + 5)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1|(-2|(3cos(2\u03c0x) - 1)| - 2) + 3| + 3) + 2)/(2(-1|(-2|(3cos(2\u03c0x) - 1)| - 2) + 3| + 3) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2sin(2\u03c0(-5(-3|x + 3| - 1) + 4) + 2) + 1) + 3| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(3|(-1sin(3\u03c0(-3|x| + 2) - 2) + 2) - 2| + 2) - 3|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(5(3|(-1|x - 2| + 3) + 1| + 3)^2 + 1(3|(-1|x - 2| + 3) + 1| + 3) - 3) + 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2sin(1\u03c0(-3|(5x + 1) + 3| + 3) - 3) - 2)^2 - 4(-2sin(1\u03c0(-3|(5x + 1) + 3| + 3) - 3) - 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-3(1cos(1\u03c0(4x^2 + 3x + 5))) + 4) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0(-3(1(-1x^2 - 3x - 5)^2 + 2(-1x^2 - 3x - 5) - 2) - 2) - 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2sin(2\u03c0((-2(-4x - 2))/(3(-4x - 2) + 1)) - 3) + 3) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|((-3(-1cos(2\u03c0(3x + 5) - 1) - 2) + 1)/(3(-1cos(2\u03c0(3x + 5) - 1) - 2) - 2)) - 2| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5((-2(-1(2|x + 3| - 1)^2 - 3(2|x + 3| - 1) + 4) + 1)/(2(-1(2|x + 3| - 1)^2 - 3(2|x + 3| - 1) + 4) - 3))^2 + 2((-2(-1(2|x + 3| - 1)^2 - 3(2|x + 3| - 1) + 4) + 1)/(2(-1(2|x + 3| - 1)^2 - 3(2|x + 3| - 1) + 4) - 3)) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(2cos(2\u03c0(1(2|x + 3| - 2) + 4) + 2) - 2)^2 + 4(2cos(2\u03c0(1(2|x + 3| - 2) + 4) + 2) - 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2sin(1\u03c0(-1(-2|x - 3| + 1)^2 + 4(-2|x - 3| + 1))) + 2)^2 - 3(2sin(1\u03c0(-1(-2|x - 3| + 1)^2 + 4(-2|x - 3| + 1))) + 2) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1cos(3\u03c0((2(1x - 5) - 3)/(-3(1x - 5) + 4))) + 3)| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-2|(-2cos(1\u03c0(4x^2 - 3x + 1) + 3) + 2) + 2| + 1)^2 + 3(-2|(-2cos(1\u03c0(4x^2 - 3x + 1) + 3) + 2) + 2| + 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(2|(2|(2cos(3\u03c0x - 3) + 1) + 3| - 3) + 2|) - 3| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-4(1cos(1\u03c0(1x + 3)) + 1) + 2)^2 + 2(-4(1cos(1\u03c0(1x + 3)) + 1) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(4(3cos(1\u03c0((-3x + 2)/(1x + 1))) + 3)^2 + 4(3cos(1\u03c0((-3x + 2)/(1x + 1))) + 3) + 4))/(3(4(3cos(1\u03c0((-3x + 2)/(1x + 1))) + 3)^2 + 4(3cos(1\u03c0((-3x + 2)/(1x + 1))) + 3) + 4) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-4(-3sin(2\u03c0(4x^2 + 5x - 1) + 2) + 1)^2 - 5) + 3)/(-2(-4(-3sin(2\u03c0(4x^2 + 5x - 1) + 2) + 1)^2 - 5) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2((-2(-3cos(2\u03c0(-2|x + 1|) + 1) - 3) - 2)/(3(-3cos(2\u03c0(-2|x + 1|) + 1) - 3) - 2)) - 3)/(-3((-2(-3cos(2\u03c0(-2|x + 1|) + 1) - 3) - 2)/(3(-3cos(2\u03c0(-2|x + 1|) + 1) - 3) - 2)) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2|(1sin(1\u03c0(2x + 4) - 3) + 2) + 1| + 1)^2 - 3(-2|(1sin(1\u03c0(2x + 4) - 3) + 2) + 1| + 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0((1(5((-2x - 1)/(-3x + 2))^2 - 3) + 3)/(-1(5((-2x - 1)/(-3x + 2))^2 - 3) + 2))) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1|(-3cos(1\u03c0((-1x + 3)/(-1x - 2)) + 2) - 2) - 1| - 2)| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-3(-2sin(2\u03c0(-4x + 3) - 1))^2 - 2(-2sin(2\u03c0(-4x + 3) - 1)) + 3) - 3)/(1(-3(-2sin(2\u03c0(-4x + 3) - 1))^2 - 2(-2sin(2\u03c0(-4x + 3) - 1)) + 3) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(5((-3(3sin(3\u03c0x - 2) - 3) + 1)/(3(3sin(3\u03c0x - 2) - 3) + 2))^2 - 4((-3(3sin(3\u03c0x - 2) - 3) + 1)/(3(3sin(3\u03c0x - 2) - 3) + 2)) - 4))/(-2(5((-3(3sin(3\u03c0x - 2) - 3) + 1)/(3(3sin(3\u03c0x - 2) - 3) + 2))^2 - 4((-3(3sin(3\u03c0x - 2) - 3) + 1)/(3(3sin(3\u03c0x - 2) - 3) + 2)) - 4) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2\u03c0(-1|(-1(2|x - 1| - 2)^2 + 4(2|x - 1| - 2) + 5) + 2| - 2)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2((1(1sin(1\u03c0(1x^2 - 4x)) - 2))/(-1(1sin(1\u03c0(1x^2 - 4x)) - 2) - 3)) + 1)/(1((1(1sin(1\u03c0(1x^2 - 4x)) - 2))/(-1(1sin(1\u03c0(1x^2 - 4x)) - 2) - 3)) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((-3(3cos(2\u03c0(-1|x + 2| - 2) + 1) + 2) - 1)/(-1(3cos(2\u03c0(-1|x + 2| - 2) + 1) + 2) - 1)) + 2| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3cos(1\u03c0(-2(-5x + 2)) + 3) + 3)^2 + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-1(-2(1|x + 1| - 2)^2 - 3(1|x + 1| - 2)) - 2)/(2(-2(1|x + 1| - 2)^2 - 3(1|x + 1| - 2)) + 2))^2 + 5((-1(-2(1|x + 1| - 2)^2 - 3(1|x + 1| - 2)) - 2)/(2(-2(1|x + 1| - 2)^2 - 3(1|x + 1| - 2)) + 2)) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-1sin(2\u03c0(-4(2|x - 1| - 3)^2 + 3(2|x - 1| - 3) + 1) + 1) - 2) - 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((-2(-3sin(3\u03c0(-1|x - 1|) + 2) - 3) + 1)/(1(-3sin(3\u03c0(-1|x - 1|) + 2) - 3) + 3))^2 + 4((-2(-3sin(3\u03c0(-1|x - 1|) + 2) - 3) + 1)/(1(-3sin(3\u03c0(-1|x - 1|) + 2) - 3) + 3)) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(2|((-3(2cos(3\u03c0x - 3) - 1) + 1)/(-1(2cos(3\u03c0x - 3) - 1) - 2)) + 3| - 1) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-3|(1|(-1cos(3\u03c0x) + 1) + 3|) - 1| + 3) + 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-4(1cos(2\u03c0(-3|x + 1| - 3) + 1) + 3) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1cos(2\u03c0(5(-5x + 4) + 5) + 3) - 2) - 1| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(2(4(3cos(3\u03c0x + 3) + 1)^2 + 1(3cos(3\u03c0x + 3) + 1) + 1)^2 + 1(4(3cos(3\u03c0x + 3) + 1)^2 + 1(3cos(3\u03c0x + 3) + 1) + 1) - 5) + 3)/(1(2(4(3cos(3\u03c0x + 3) + 1)^2 + 1(3cos(3\u03c0x + 3) + 1) + 1)^2 + 1(4(3cos(3\u03c0x + 3) + 1)^2 + 1(3cos(3\u03c0x + 3) + 1) + 1) - 5) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1(1(-1sin(3\u03c0x - 2) + 3))^2 - 5(1(-1sin(3\u03c0x - 2) + 3)) - 4) + 1)/(1(-1(1(-1sin(3\u03c0x - 2) + 3))^2 - 5(1(-1sin(3\u03c0x - 2) + 3)) - 4)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0(-4((-2(-4x - 4) - 1)/(2(-4x - 4) - 3))^2 + 3) + 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(3(3cos(1\u03c0(-3|x| + 1) - 1) + 2) + 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2sin(2\u03c0(1|(-3x + 3) - 3| + 1) + 3) + 1) - 1| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4((-1(-1cos(2\u03c0(2x - 4) + 2) - 2))/(-2(-1cos(2\u03c0(2x - 4) + 2) - 2) - 3)) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-5(1cos(3\u03c0(-2|x + 1| - 1) + 3) - 3)^2 - 5(1cos(3\u03c0(-2|x + 1| - 1) + 3) - 3) - 2) + 3)/(-3(-5(1cos(3\u03c0(-2|x + 1| - 1) + 3) - 3)^2 - 5(1cos(3\u03c0(-2|x + 1| - 1) + 3) - 3) - 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((-3(-3|(-1sin(2\u03c0x - 3) - 1) - 1| + 3) + 3)/(1(-3|(-1sin(2\u03c0x - 3) - 1) - 1| + 3))) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-3sin(3\u03c0(4(2x - 5)^2 - 5(2x - 5) - 5)) - 2) + 1| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-3cos(2\u03c0(-3((2x)/(-2x - 3)) + 1) + 1) + 3) + 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(2(1|(4x^2 - 5) + 2| + 3) - 4) - 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-1|(1sin(3\u03c0(3x^2 + 4x - 1) - 1) + 2) + 2| + 3) - 2)/(3(-1|(1sin(3\u03c0(3x^2 + 4x - 1) - 1) + 2) + 2| + 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1cos(3\u03c0(-4(-1|x| + 3)^2 + 1(-1|x| + 3) - 4) + 1) - 2) - 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2|(1sin(3\u03c0(-2|x + 1| + 3) + 1)) - 2| - 2)^2 + 2(-2|(1sin(3\u03c0(-2|x + 1| + 3) + 1)) - 2| - 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(1(1cos(2\u03c0(1x) - 3) - 1) - 3) - 1| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(2|(2|(2sin(2\u03c0x - 3) + 2) + 2| - 2) + 3|) - 1| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1(2cos(3\u03c0(-1x^2 - 5x - 2) - 3) - 3)^2 - 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-2cos(1\u03c0(-3|((-3x)/(-2x + 3))| + 1) + 1))^2 + 2(-2cos(1\u03c0(-3|((-3x)/(-2x + 3))| + 1) + 1)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((2(1cos(2\u03c0((2x - 2)/(2x - 1)) - 1) + 1) - 1)/(3(1cos(2\u03c0((2x - 2)/(2x - 1)) - 1) + 1) - 2))^2 - 5((2(1cos(2\u03c0((2x - 2)/(2x - 1)) - 1) + 1) - 1)/(3(1cos(2\u03c0((2x - 2)/(2x - 1)) - 1) + 1) - 2)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(4(-3cos(3\u03c0(1|x + 2| + 1) - 2) + 2) - 3)^2 - 2(4(-3cos(3\u03c0(1|x + 2| + 1) - 2) + 2) - 3) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0(-5((-2(1x + 2) + 3)/(-2(1x + 2) + 1))^2 - 1((-2(1x + 2) + 3)/(-2(1x + 2) + 1)) + 5) - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((-1(-2(-1sin(1\u03c0x) - 3) - 2) + 3)/(-3(-2(-1sin(1\u03c0x) - 3) - 2) + 2))| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(3|((-3(-3sin(3\u03c0x + 1) - 2))/(-3(-3sin(3\u03c0x + 1) - 2) - 3)) + 1| + 3)^2 - 2(3|((-3(-3sin(3\u03c0x + 1) - 2))/(-3(-3sin(3\u03c0x + 1) - 2) - 3)) + 1| + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(1cos(3\u03c0(1|(-4x) - 3| - 3)) + 2) + 2)/(-1(1cos(3\u03c0(1|(-4x) - 3| - 3)) + 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(2sin(3\u03c0(1(-1x + 4)^2 + 2(-1x + 4) - 3) - 3)) + 1)/(-3(2sin(3\u03c0(1(-1x + 4)^2 + 2(-1x + 4) - 3) - 3)) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2cos(3\u03c0(3(4x - 3)^2 + 4(4x - 3) + 2) - 1) - 1) + 3)/(-3(-2cos(3\u03c0(3(4x - 3)^2 + 4(4x - 3) + 2) - 1) - 1) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(-3((-2((3x)/(2x - 1)) + 1)/(1((3x)/(2x - 1)) - 2))^2 - 4((-2((3x)/(2x - 1)) + 1)/(1((3x)/(2x - 1)) - 2)) + 5) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2((-3(1cos(1\u03c0((3x - 2)/(1x + 2)) - 3) + 3) - 1)/(-2(1cos(1\u03c0((3x - 2)/(1x + 2)) - 3) + 3) + 3)) + 1)/(3((-3(1cos(1\u03c0((3x - 2)/(1x + 2)) - 3) + 3) - 1)/(-2(1cos(1\u03c0((3x - 2)/(1x + 2)) - 3) + 3) + 3)) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2|(-2|(3|x + 3|) - 2|) + 3| + 1)^2 + 1(-2|(-2|(3|x + 3|) - 2|) + 3| + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2cos(2\u03c0(-3(-2|x + 2| + 3) + 1) + 3) - 3)^2 - 1(-2cos(2\u03c0(-3(-2|x + 2| + 3) + 1) + 3) - 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3(-1(5x^2 + 3x)^2 + 5(5x^2 + 3x) - 3) + 2)/(1(-1(5x^2 + 3x)^2 + 5(5x^2 + 3x) - 3) + 3))^2 + 5((-3(-1(5x^2 + 3x)^2 + 5(5x^2 + 3x) - 3) + 2)/(1(-1(5x^2 + 3x)^2 + 5(5x^2 + 3x) - 3) + 3)) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3cos(3\u03c0(3(-3|x + 3| - 1) + 2) - 1) - 1) - 2)/(-1(-3cos(3\u03c0(3(-3|x + 3| - 1) + 2) - 1) - 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-1|(2(3cos(3\u03c0x + 2) + 2)^2 - 5(3cos(3\u03c0x + 2) + 2) - 4) - 2| - 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(3sin(2\u03c0(-1|(3|x - 2| + 2) - 3| - 2) + 1) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(-2|(-2|((3x + 2)/(-2x + 3)) + 3| - 1) + 1|) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-3cos(3\u03c0((-2(-5x + 2) - 3)/(1(-5x + 2))) - 2)) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1|(-2(3|x|)) + 2| - 2)^2 - 2(1|(-2(3|x|)) + 2| - 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2((-1(-2cos(2\u03c0x + 2) - 3) + 3)/(2(-2cos(2\u03c0x + 2) - 3) + 3)) - 2)^2 + 5(2((-1(-2cos(2\u03c0x + 2) - 3) + 3)/(2(-2cos(2\u03c0x + 2) - 3) + 3)) - 2) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(3|(2|(-2x^2 - 5x + 2) + 1| - 2) - 1| - 1) + 3| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1cos(3\u03c0((-1(4x + 5) - 1)/(1(4x + 5) - 3)) - 3) + 1)| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(-1|(1(3|x + 1|) - 1) - 1| + 2) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|((-1(1(-1cos(3\u03c0x + 2) + 3)^2 + 2(-1cos(3\u03c0x + 2) + 3) + 1) + 1)/(2(1(-1cos(3\u03c0x + 2) + 3)^2 + 2(-1cos(3\u03c0x + 2) + 3) + 1) - 1))| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1|(4(1sin(3\u03c0x + 3) + 1))| - 1)| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3\u03c0(-1((-3((1x - 3)/(1x - 2)) - 1)/(-2((1x - 3)/(1x - 2)) + 3)) - 3) + 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4((-2(-3cos(1\u03c0(-2x - 2) - 3) - 2) - 1)/(2(-3cos(1\u03c0(-2x - 2) - 3) - 2)))^2 - 4((-2(-3cos(1\u03c0(-2x - 2) - 3) - 2) - 1)/(2(-3cos(1\u03c0(-2x - 2) - 3) - 2))) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-5(-2sin(2\u03c0(-2|x - 2| + 3) - 3) - 1)^2 + 2(-2sin(2\u03c0(-2|x - 2| + 3) - 3) - 1) + 1) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-3|(4(-2cos(3\u03c0x + 3) + 3)^2 - 3(-2cos(3\u03c0x + 3) + 3) - 1)| - 1) - 1| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-5(1((1x - 3)/(-2x + 3)) + 4) + 5) + 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-5(-3(-2sin(1\u03c0x + 1)) - 3)^2 - 1) - 2)/(-3(-5(-3(-2sin(1\u03c0x + 1)) - 3)^2 - 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(5(2|(-4x^2 - 2x - 2) + 3| - 2) + 3) - 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3|(2|(1sin(3\u03c0x + 3) + 3) - 3| - 3)| + 3)| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((3(2sin(3\u03c0((1x - 1)/(2x - 3)) + 3) + 3) + 3)/(-1(2sin(3\u03c0((1x - 1)/(2x - 3)) + 3) + 3))) + 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(2sin(3\u03c0((2(-2x - 2) + 1)/(-2(-2x - 2) - 3)) + 3) + 1) + 2| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(3|(1cos(3\u03c0(-2x^2 - 5x + 3) + 1) - 2) - 2| + 3) - 1)/(1(3|(1cos(3\u03c0(-2x^2 - 5x + 3) + 1) - 2) - 2| + 3) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-3(-1cos(2\u03c0(-5x - 2) - 1) - 1)^2 + 4(-1cos(2\u03c0(-5x - 2) - 1) - 1) + 4) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(-3(-2(-1|x - 2| - 1) - 4) + 4) + 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3\u03c0(-1|(-3(-2x + 5)^2 + 5(-2x + 5) + 2)| + 3) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1(-2|(3x^2 - 3) - 1| + 2)^2 - 3(-2|(3x^2 - 3) - 1| + 2) + 5)^2 - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(2((2(-2cos(2\u03c0x - 1) + 3) - 3)/(2(-2cos(2\u03c0x - 1) + 3))) + 1) + 1)/(3(2((2(-2cos(2\u03c0x - 1) + 3) - 3)/(2(-2cos(2\u03c0x - 1) + 3))) + 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(1sin(1\u03c0(4((2x - 1)/(-1x - 3))^2 + 4((2x - 1)/(-1x - 3)) + 4) + 2) - 3) + 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2|(1sin(3\u03c0(2|x + 3| - 2) - 1) + 3) - 3| - 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(4(3cos(3\u03c0(3|x + 3| + 2) + 2) + 3) + 5) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-1cos(1\u03c0(-2|(1x^2 + 4x - 2) - 3| + 3) + 2) + 3)^2 + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1(-2sin(3\u03c0(-3x - 1)) - 2) - 4)^2 - 2(1(-2sin(3\u03c0(-3x - 1)) - 2) - 4) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2cos(3\u03c0((2(1|x + 2|) + 1)/(-3(1|x + 2|) + 2)) - 2)))/(-3(-2cos(3\u03c0((2(1|x + 2|) + 1)/(-3(1|x + 2|) + 2)) - 2)) + 4) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3cos(3\u03c0(2(-3|x - 2| + 2) + 4) - 1) - 3) + 2| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-2sin(2\u03c0(-5(3x + 4) - 5)) + 1) + 2)/(-1(-2sin(2\u03c0(-5(3x + 4) - 5)) + 1) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "105", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1sin(1\u03c0(-5(1|x - 1| - 3)^2 + 2) - 3) - 3) + 2| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2\u03c0(-4(-2|((1x - 1)/(-3x - 2))|)^2 + 5(-2|((1x - 1)/(-3x - 2))|) + 2) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2(2sin(3\u03c0(-5x^2 + 1x + 4) + 3) + 1) + 1) + 2)/(2(-2(2sin(3\u03c0(-5x^2 + 1x + 4) + 3) + 1) + 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|((-2(-3(-1cos(2\u03c0x - 3) + 1) - 4) + 1)/(1(-3(-1cos(2\u03c0x - 3) + 1) - 4))) + 1| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1(-1|(1sin(2\u03c0x - 3) - 2) + 3| + 1)^2 + 2(-1|(1sin(2\u03c0x - 3) - 2) + 3| + 1) - 1)^2 + 2(-1(-1|(1sin(2\u03c0x - 3) - 2) + 3| + 1)^2 + 2(-1|(1sin(2\u03c0x - 3) - 2) + 3| + 1) - 1) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-2cos(3\u03c0(-5(1x - 5)^2 - 1(1x - 5) - 4) - 3) + 3) - 2)/(1(-2cos(3\u03c0(-5(1x - 5)^2 - 1(1x - 5) - 4) - 3) + 3) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4((-2(-2cos(3\u03c0(-5x - 4) - 2) + 1))/(-3(-2cos(3\u03c0(-5x - 4) - 2) + 1) - 3)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2cos(1\u03c0(3|(1x - 2) - 2| - 1) + 1) - 3)^2 - 5(-2cos(1\u03c0(3|(1x - 2) - 2| - 1) + 1) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-2cos(1\u03c0((1((3x)/(2x + 3)) + 2)/(3((3x)/(2x + 3)))) - 1) - 3))/(2(-2cos(1\u03c0((1((3x)/(2x + 3)) + 2)/(3((3x)/(2x + 3)))) - 1) - 3) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1(-1(3sin(2\u03c0x + 1) + 2))^2 + 4(-1(3sin(2\u03c0x + 1) + 2)) - 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0(2(-4((3x - 2)/(1x + 1))^2 + 3((3x - 2)/(1x + 1)) - 3) - 2)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(3|(-4((-2x + 1)/(2x + 1))^2 + 4((-2x + 1)/(2x + 1)) + 2) + 2| - 2) + 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1(2|(3cos(2\u03c0x - 1))| - 3) + 2)^2 + 5(1(2|(3cos(2\u03c0x - 1))| - 3) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4((-1(2sin(1\u03c0(-4x^2 - 5) + 3)) + 3)/(-2(2sin(1\u03c0(-4x^2 - 5) + 3)))) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2sin(2\u03c0(2|(1|x - 3| + 3) - 1|)) + 2) - 2)/(-2(2sin(2\u03c0(2|(1|x - 3| + 3) - 1|)) + 2) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(1(1cos(2\u03c0((-2x + 3)/(2x + 3)) - 2) - 2) - 5)^2 - 1(1(1cos(2\u03c0((-2x + 3)/(2x + 3)) - 2) - 2) - 5) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1(3cos(2\u03c0((-2x - 3)/(2x - 1)) + 3) - 2)^2 - 5(3cos(2\u03c0((-2x - 3)/(2x - 1)) + 3) - 2)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-3cos(1\u03c0(5(2|x + 2| - 2)^2 - 3(2|x + 2| - 2) + 2) + 3) - 3))/(3(-3cos(1\u03c0(5(2|x + 2| - 2)^2 - 3(2|x + 2| - 2) + 2) + 3) - 3) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(-1|(-3(-5x - 1)^2 + 4(-5x - 1) + 3) + 3| - 1)) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(1(-2|(-3|x + 1| + 1) + 2| - 2) + 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-2|(2|(-3sin(3\u03c0x + 1) - 2) - 1| - 2)| + 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(1\u03c0((-2(4((-1x + 2)/(-3x - 1))^2 + 1((-1x + 2)/(-3x - 1)) - 5) + 3)/(2(4((-1x + 2)/(-3x - 1))^2 + 1((-1x + 2)/(-3x - 1)) - 5)))) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-2|((2(-1|x - 1| + 2) + 1)/(-1(-1|x - 1| + 2) - 3)) - 1| + 3)^2 + 1(-2|((2(-1|x - 1| + 2) + 1)/(-1(-1|x - 1| + 2) - 3)) - 1| + 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-1(-2(2sin(1\u03c0x + 1) - 2)^2 - 2(2sin(1\u03c0x + 1) - 2) - 3) + 3)/(-2(-2(2sin(1\u03c0x + 1) - 2)^2 - 2(2sin(1\u03c0x + 1) - 2) - 3)))^2 + 4((-1(-2(2sin(1\u03c0x + 1) - 2)^2 - 2(2sin(1\u03c0x + 1) - 2) - 3) + 3)/(-2(-2(2sin(1\u03c0x + 1) - 2)^2 - 2(2sin(1\u03c0x + 1) - 2) - 3))) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1cos(1\u03c0(-1((3x + 3)/(-1x + 3))^2 - 4((3x + 3)/(-1x + 3)) - 3)) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-1|(-1(-2sin(1\u03c0x))^2 + 4(-2sin(1\u03c0x))) + 2| - 3)^2 + 1(-1|(-1(-2sin(1\u03c0x))^2 + 4(-2sin(1\u03c0x))) + 2| - 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(-1((2((2x - 3)/(-3x - 1)) - 2)/(3((2x - 3)/(-3x - 1)) - 1))) - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(2(2cos(3\u03c0(3x + 2) - 2) - 1)^2 + 4(2cos(3\u03c0(3x + 2) - 2) - 1) - 3) + 2)/(2(2(2cos(3\u03c0(3x + 2) - 2) - 1)^2 + 4(2cos(3\u03c0(3x + 2) - 2) - 1) - 3) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3sin(3\u03c0(2(-3x^2 - 3x - 1) - 4) - 2)) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3|(-3cos(3\u03c0(1x^2 - 1x + 1) + 2)) - 3| - 3)^2 + 1(3|(-3cos(3\u03c0(1x^2 - 1x + 1) + 2)) - 3| - 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(-4((1(-1x - 4) - 2)/(-2(-1x - 4) + 3)) + 1) + 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1((-1(2|(2cos(2\u03c0x + 1) - 1) - 3| - 3) + 3)/(-1(2|(2cos(2\u03c0x + 1) - 1) - 3| - 3) + 2)) - 2)/(1((-1(2|(2cos(2\u03c0x + 1) - 1) - 3| - 3) + 3)/(-1(2|(2cos(2\u03c0x + 1) - 1) - 3| - 3) + 2)) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2(-3cos(1\u03c0(3x^2 + 3x + 2) + 3) + 3)^2 + 4(-3cos(1\u03c0(3x^2 + 3x + 2) + 3) + 3) + 4) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-2((-1(3|x| - 3) - 3)/(-2(3|x| - 3) + 3))^2 + 2((-1(3|x| - 3) - 3)/(-2(3|x| - 3) + 3)) - 1) + 2| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-5(-1|((2x + 2)/(-2x)) + 2| + 3)^2 - 4(-1|((2x + 2)/(-2x)) + 2| + 3) + 1) + 2| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0((-1(-3(-1x^2 + 4x - 1)^2 + 1(-1x^2 + 4x - 1) + 4) - 2)/(-3(-3(-1x^2 + 4x - 1)^2 + 1(-1x^2 + 4x - 1) + 4) - 2)) + 1) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3|(1cos(1\u03c0(4x + 1) + 2) + 3) + 1| + 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-1(2(1cos(1\u03c0x + 2) + 2) + 5) - 4)^2 + 1(-1(2(1cos(1\u03c0x + 2) + 2) + 5) - 4) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0((-1(-1((-1x + 2)/(-1x - 1)) - 2) + 3)/(1(-1((-1x + 2)/(-1x - 1)) - 2))) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2cos(3\u03c0((-2((2x - 3)/(2x - 1)) - 1)/(-1((2x - 3)/(2x - 1)) + 2)) + 3) + 2) + 2| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-4(-1sin(3\u03c0(5x^2 + 5x + 2) + 2) + 2) - 1) - 2| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(1|(-5(-2x + 5) + 2)|)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1((-2(2(-2sin(3\u03c0x - 2) - 3)^2 - 5))/(-3(2(-2sin(3\u03c0x - 2) - 3)^2 - 5) + 4)) + 3)/(2((-2(2(-2sin(3\u03c0x - 2) - 3)^2 - 5))/(-3(2(-2sin(3\u03c0x - 2) - 3)^2 - 5) + 4)) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0((-1(-1(-5x^2 - 1x)^2 - 4(-5x^2 - 1x) + 3) - 3)/(3(-1(-5x^2 - 1x)^2 - 4(-5x^2 - 1x) + 3) - 2)) + 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-3|((-1(-3sin(1\u03c0x - 1) + 1) + 1)/(-1(-3sin(1\u03c0x - 1) + 1) + 2)) + 1| - 2))/(-2(-3|((-1(-3sin(1\u03c0x - 1) + 1) + 1)/(-1(-3sin(1\u03c0x - 1) + 1) + 2)) + 1| - 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-3cos(1\u03c0(-5(-1|x| + 3)) + 2) + 1) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-1sin(2\u03c0(2((3x)/(-1x + 3))^2 - 1((3x)/(-1x + 3)) + 5) + 3) - 1)^2 + 4(-1sin(2\u03c0(2((3x)/(-1x + 3))^2 - 1((3x)/(-1x + 3)) + 5) + 3) - 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(1(-2cos(2\u03c0((3x)/(-2x - 2)) + 3) - 3)^2 - 5(-2cos(2\u03c0((3x)/(-2x - 2)) + 3) - 3) - 4) - 2)/(2(1(-2cos(2\u03c0((3x)/(-2x - 2)) + 3) - 3)^2 - 5(-2cos(2\u03c0((3x)/(-2x - 2)) + 3) - 3) - 4) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(3(1cos(3\u03c0((-1x - 2)/(2x - 1)) - 2) - 2)^2 + 5) - 3)/(-2(3(1cos(3\u03c0((-1x - 2)/(2x - 1)) - 2) - 2)^2 + 5) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(1\u03c0(-2|(-2(3|x + 1| - 2)^2 + 5(3|x + 1| - 2) - 3) - 3| + 3)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(-5(-3|((2x - 1)/(2x)) - 3|) - 2) - 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0((3((-1(-2x^2 + 3x + 5) + 2)/(3(-2x^2 + 3x + 5) + 3)))/(3((-1(-2x^2 + 3x + 5) + 2)/(3(-2x^2 + 3x + 5) + 3)) - 1)) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0((-1(-4(1|x + 2|) + 1) - 3)/(-3(-4(1|x + 2|) + 1))) - 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(-3|(-3(1x + 3)^2 - 1(1x + 3) + 5) + 1| - 3) - 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2(1cos(1\u03c0(3x^2 + 4x + 4) - 2) + 3) - 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3cos(2\u03c0((-1(2x^2 - 5x - 2) - 3)/(1(2x^2 - 5x - 2))) - 2) - 2)^2 + 5(3cos(2\u03c0((-1(2x^2 - 5x - 2) - 3)/(1(2x^2 - 5x - 2))) - 2) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1cos(3\u03c0(5((-3x + 1)/(2x)) + 2)))^2 - 4(1cos(3\u03c0(5((-3x + 1)/(2x)) + 2))) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(2\u03c0((-1(-2|(-1|x + 1| + 3)| + 3) + 3)/(-1(-2|(-1|x + 1| + 3)| + 3) + 2)) + 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2((-2(1sin(3\u03c0((-2x + 2)/(-2x)) + 1) - 3) - 2)/(-2(1sin(3\u03c0((-2x + 2)/(-2x)) + 1) - 3))) + 3)/(1((-2(1sin(3\u03c0((-2x + 2)/(-2x)) + 1) - 3) - 2)/(-2(1sin(3\u03c0((-2x + 2)/(-2x)) + 1) - 3))) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(1\u03c0(-2|(-3(-2x + 2)^2 + 4(-2x + 2) + 3)| + 1) - 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(3(1cos(3\u03c0(3x^2 + 2x + 4) + 2) - 2)^2 - 1(1cos(3\u03c0(3x^2 + 2x + 4) + 2) - 2) - 1) + 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(5(-4((-1x)/(-3x + 2))^2 - 3((-1x)/(-3x + 2)) - 2)^2 - 5) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1((-3(-2|(2cos(1\u03c0x + 3) - 2) + 2| - 1))/(3(-2|(2cos(1\u03c0x + 3) - 2) + 2| - 1) + 2)) - 3)/(1((-3(-2|(2cos(1\u03c0x + 3) - 2) + 2| - 1))/(3(-2|(2cos(1\u03c0x + 3) - 2) + 2| - 1) + 2))) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((2((1(-1sin(2\u03c0x) - 1) + 2)/(-3(-1sin(2\u03c0x) - 1) - 3)) + 1)/(-3((1(-1sin(2\u03c0x) - 1) + 2)/(-3(-1sin(2\u03c0x) - 1) - 3)) - 1)) - 2| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-3cos(2\u03c0(3|(-2x - 2) - 3| - 1) - 1) + 3)| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(1sin(2\u03c0(-2(3|x + 2|)^2 - 4(3|x + 2|) + 4) + 2) + 2) - 1)/(1(1sin(2\u03c0(-2(3|x + 2|)^2 - 4(3|x + 2|) + 4) + 2) + 2) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((3((-3(3cos(1\u03c0x - 1) + 1) - 3)/(-2(3cos(1\u03c0x - 1) + 1) + 3)) + 3)/(1((-3(3cos(1\u03c0x - 1) + 1) - 3)/(-2(3cos(1\u03c0x - 1) + 1) + 3))))^2 - 3((3((-3(3cos(1\u03c0x - 1) + 1) - 3)/(-2(3cos(1\u03c0x - 1) + 1) + 3)) + 3)/(1((-3(3cos(1\u03c0x - 1) + 1) - 3)/(-2(3cos(1\u03c0x - 1) + 1) + 3)))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1((-2(-3(1cos(2\u03c0x) - 2)^2 + 3(1cos(2\u03c0x) - 2) + 3) + 1)/(-3(-3(1cos(2\u03c0x) - 2)^2 + 3(1cos(2\u03c0x) - 2) + 3) - 1)) - 2)/(-3((-2(-3(1cos(2\u03c0x) - 2)^2 + 3(1cos(2\u03c0x) - 2) + 3) + 1)/(-3(-3(1cos(2\u03c0x) - 2)^2 + 3(1cos(2\u03c0x) - 2) + 3) - 1)) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4((-3(-5(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) + 3) + 1)/(-2(-5(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) + 3) - 3))^2 + 5((-3(-5(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) + 3) + 1)/(-2(-5(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) + 3) - 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(-3|(1((3x - 2)/(-3x))^2 + 2((3x - 2)/(-3x)) - 5) + 1| - 3) + 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-1cos(1\u03c0(-3(3x - 3)^2 - 3(3x - 3) - 3) - 3) - 1) - 3| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-5(2sin(3\u03c0(2|x + 3| - 3) + 1) + 3)^2 - 4(2sin(3\u03c0(2|x + 3| - 3) + 1) + 3) - 4) + 2| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-2cos(2\u03c0(1(1|x - 3|)^2 + 5) - 2) + 1) + 2)/(2(-2cos(2\u03c0(1(1|x - 3|)^2 + 5) - 2) + 1) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(3(1|(1cos(3\u03c0x - 2) + 1) - 3| + 3) + 5)^2 + 1(3(1|(1cos(3\u03c0x - 2) + 1) - 3| + 3) + 5) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3|(2cos(2\u03c0(3x^2 - 3x + 1) + 1) - 2) + 2| - 3)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(5((1(-3cos(1\u03c0x) + 3) - 1)/(1(-3cos(1\u03c0x) + 3) + 3))^2 - 4((1(-3cos(1\u03c0x) + 3) - 1)/(1(-3cos(1\u03c0x) + 3) + 3)) + 5) + 2| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(-3|(4(-3x) - 5)| + 3) + 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1sin(3\u03c0((2(-1|x + 1| - 2) + 3)/(-1(-1|x + 1| - 2) + 2)) + 3) - 3)^2 - 5(-1sin(3\u03c0((2(-1|x + 1| - 2) + 3)/(-1(-1|x + 1| - 2) + 2)) + 3) - 3) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-5(1sin(1\u03c0(-3|x + 3| - 1) - 3))^2 - 1(1sin(1\u03c0(-3|x + 3| - 1) - 3)) + 4) + 2)/(1(-5(1sin(1\u03c0(-3|x + 3| - 1) - 3))^2 - 1(1sin(1\u03c0(-3|x + 3| - 1) - 3)) + 4) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-2(-3|(-1sin(3\u03c0x - 1) - 2) + 2| - 1) - 4) - 2|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(3sin(2\u03c0(3|(1x - 4) - 2| - 3) - 3) - 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|((-3(3cos(3\u03c0(-2|x - 1| + 1) - 3) - 1) - 3)/(-2(3cos(3\u03c0(-2|x - 1| + 1) - 3) - 1) + 3))| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(1(1(1x^2 + 2x + 3)^2 + 1(1x^2 + 2x + 3) - 1)^2 - 2(1(1x^2 + 2x + 3)^2 + 1(1x^2 + 2x + 3) - 1) + 4) + 1) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-2sin(2\u03c0(1((-3x)/(1x + 2)) + 5) - 3)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2((-2(2cos(2\u03c0x - 3) - 3) + 2)/(-1(2cos(2\u03c0x - 3) - 3) - 1))^2 - 4((-2(2cos(2\u03c0x - 3) - 3) + 2)/(-1(2cos(2\u03c0x - 3) - 3) - 1)) + 2)^2 + 3(2((-2(2cos(2\u03c0x - 3) - 3) + 2)/(-1(2cos(2\u03c0x - 3) - 3) - 1))^2 - 4((-2(2cos(2\u03c0x - 3) - 3) + 2)/(-1(2cos(2\u03c0x - 3) - 3) - 1)) + 2) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1cos(2\u03c0((-3(-1x - 4) + 2)/(3(-1x - 4) + 1)) + 1) + 3)^2 - 2(-1cos(2\u03c0((-3(-1x - 4) + 2)/(3(-1x - 4) + 1)) + 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3|(3sin(2\u03c0(-1x) - 2) - 2) - 2| + 1) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(2e^(1(1|(1x^2 + 4x + 1) - 2| - 3)) - 2) - 1| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1cos(3\u03c0(1|(1x + 5) + 3| - 1) - 3) - 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(3|(-3|(3x^2 + 1x - 2) + 1| - 3) + 2| + 1) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0(5(-4(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 2)^2 - 2(-4(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 2) - 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(1(1((-2x + 2)/(1x + 2)) - 4) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3sin(3\u03c0(2(5x + 5) + 2) + 3) - 3) - 1)/(1(3sin(3\u03c0(2(5x + 5) + 2) + 3) - 3)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0((2(-2(-3x - 3) - 4) - 3)/(2(-2(-3x - 3) - 4) + 2))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(1(1|(-1sin(3\u03c0x + 2) + 2) - 3|)^2 + 4(1|(-1sin(3\u03c0x + 2) + 2) - 3|) - 2)| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(3((2(-5x - 5) + 2)/(1(-5x - 5)))^2 - 1((2(-5x - 5) + 2)/(1(-5x - 5))) - 3) - 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-1|(-3cos(2\u03c0(-5x - 1) - 2) - 3)|) - 1)/(-3(-1|(-3cos(2\u03c0(-5x - 1) - 2) - 3)|) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-3|(2cos(3\u03c0((-2x - 3)/(-1x + 2))) - 3) - 1| + 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-5(2cos(2\u03c0(-3|x| + 3) - 1) + 1)^2 - 4(2cos(2\u03c0(-3|x| + 3) - 1) + 1) + 5) + 1)/(3(-5(2cos(2\u03c0(-3|x| + 3) - 1) + 1)^2 - 4(2cos(2\u03c0(-3|x| + 3) - 1) + 1) + 5) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1cos(1\u03c0(4(-4x - 1)^2 + 1(-4x - 1) + 2) + 3) - 1)^2 - 2(1cos(1\u03c0(4(-4x - 1)^2 + 1(-4x - 1) + 2) + 3) - 1) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(5((-2(3|x + 2| - 3) - 3)/(-1(3|x + 2| - 3) + 3))^2 + 1((-2(3|x + 2| - 3) - 3)/(-1(3|x + 2| - 3) + 3)) + 1)^2 + 3(5((-2(3|x + 2| - 3) - 3)/(-1(3|x + 2| - 3) + 3))^2 + 1((-2(3|x + 2| - 3) - 3)/(-1(3|x + 2| - 3) + 3)) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0((-2(2|(2|x + 3| + 2) - 2| - 2) - 1)/(-1(2|(2|x + 3| + 2) - 2| - 2))) + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3\u03c0(4(1|(4x - 5) - 3| + 1) - 2)) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0((1((1(1x^2 - 4x) + 1)/(3(1x^2 - 4x) - 2)) + 2)/(-3((1(1x^2 - 4x) + 1)/(3(1x^2 - 4x) - 2)) + 1)) - 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-2|(3|(3cos(2\u03c0x + 3) - 3) + 1| + 2) + 3| + 3) + 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(1(1|(2sin(1\u03c0x) - 1) - 3|) - 1) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-4(2(3sin(1\u03c0x + 1) + 1)^2 - 4(3sin(1\u03c0x + 1) + 1) + 5)^2 - 1(2(3sin(1\u03c0x + 1) + 1)^2 - 4(3sin(1\u03c0x + 1) + 1) + 5) + 3) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(1sin(2\u03c0((2(-4x^2 - 3x) - 3)/(-3(-4x^2 - 3x) + 2)) + 3) + 2))/(-1(1sin(2\u03c0((2(-4x^2 - 3x) - 3)/(-3(-4x^2 - 3x) + 2)) + 3) + 2) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(2cos(3\u03c0(-3(3|x + 1| + 3) + 1)) - 2)^2 + 4(2cos(3\u03c0(-3(3|x + 1| + 3) + 1)) - 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1cos(1\u03c0(5(-3|x - 2| - 2)^2 - 5(-3|x - 2| - 2) - 1) + 2) - 2) + 3| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(3|((2(3sin(3\u03c0x + 2) + 2) - 2)/(3(3sin(3\u03c0x + 2) + 2) - 2))| + 1) + 3)/(3(3|((2(3sin(3\u03c0x + 2) + 2) - 2)/(3(3sin(3\u03c0x + 2) + 2) - 2))| + 1) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1sin(1\u03c0(3|(4x^2 - 5x - 1) + 2|)) + 3)^2 - 3(-1sin(1\u03c0(3|(4x^2 - 5x - 1) + 2|)) + 3) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0((3(-2|((-3x + 1)/(3x)) + 3| + 3) - 2)/(1(-2|((-3x + 1)/(3x)) + 3| + 3) + 1))) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3|(-1sin(1\u03c0(2|x + 1| + 2) - 3) + 1) - 2| - 2) - 1)/(-2(3|(-1sin(1\u03c0(2|x + 1| + 2) - 3) + 1) - 2| - 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0((-1(2((-2x + 2)/(2x + 3))^2 - 4((-2x + 2)/(2x + 3)) - 5) - 2)/(1(2((-2x + 2)/(2x + 3))^2 - 4((-2x + 2)/(2x + 3)) - 5) + 1))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1e^(-1(-1(2|(4x^2 - 2x - 1) - 2| - 1)^2 + 5(2|(4x^2 - 2x - 1) - 2| - 1) - 4)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(2|(-3(-1|x - 3| - 2)^2 - 5(-1|x - 3| - 2) + 3) + 2| + 2) + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2|(-5(2cos(1\u03c0x - 3) - 3)) + 3| - 1) + 2)/(-3(-2|(-5(2cos(1\u03c0x - 3) - 3)) + 3| - 1) + 4), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(5(-5((-1x + 1)/(-1x + 3)) + 5)^2 + 3(-5((-1x + 1)/(-1x + 3)) + 5) + 4) - 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-2cos(2\u03c0(5((-1x + 1)/(-2x))^2 + 4((-1x + 1)/(-2x)) + 5) + 2) - 2) - 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1cos(2\u03c0((-3(2x - 4) - 1)/(1(2x - 4) + 3)) - 1) + 1)^2 + 3(1cos(2\u03c0((-3(2x - 4) - 1)/(1(2x - 4) + 3)) - 1) + 1) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(1sin(2\u03c0(-2(1x - 4) - 4) + 1) - 3)^2 - 1(1sin(2\u03c0(-2(1x - 4) - 4) + 1) - 3) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2(1cos(1\u03c0(-1x^2 + 1x + 1) - 3) - 2) + 4)^2 - 2(-2(1cos(1\u03c0(-1x^2 + 1x + 1) - 3) - 2) + 4) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(3cos(3\u03c0((3(-5x + 4) - 1)/(-2(-5x + 4) + 3)) + 2) + 1) - 1| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(2(4(-1sin(2\u03c0x - 2) - 1) - 4) + 1) + 3| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3\u03c0(-1|(1|(-1|x + 2| + 1)| - 2) + 1| + 2) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1(3cos(1\u03c0(2|x| + 3) + 1) + 2)^2 + 4(3cos(1\u03c0(2|x| + 3) + 1) + 2) - 2) - 2| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-4(-5(-3cos(2\u03c0x + 2) + 3)^2 + 2(-3cos(2\u03c0x + 2) + 3) - 2) + 2) + 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(3((-1(-5x + 1))/(-1(-5x + 1) + 2))^2 - 3((-1(-5x + 1))/(-1(-5x + 1) + 2)) - 2) + 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(1(2(3x - 2) - 1)^2 + 1(2(3x - 2) - 1) - 4) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-2|((3(-1sin(1\u03c0x - 3) + 2) + 2)/(-2(-1sin(1\u03c0x - 3) + 2) + 3)) - 3| + 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2cos(3\u03c0((1(1x) + 3)/(-3(1x) - 1)) + 1) + 3) - 1)/(2(-2cos(3\u03c0((1(1x) + 3)/(-3(1x) - 1)) + 1) + 3) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(2cos(1\u03c0(-1|((2x + 3)/(1x + 2)) - 2| + 3) - 3) + 1) - 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1sin(2\u03c0(-1(5x^2 + 5x + 2)^2 + 3(5x^2 + 5x + 2)) + 2) - 2)^2 - 2(1sin(2\u03c0(-1(5x^2 + 5x + 2)^2 + 3(5x^2 + 5x + 2)) + 2) - 2) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(2|(1(-1x^2 - 5x + 1)^2 + 2(-1x^2 - 5x + 1) - 1) - 2| + 1) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2sin(3\u03c0(-1(3x + 4)^2 + 3(3x + 4) - 3) - 1) - 1)^2 + 4(2sin(3\u03c0(-1(3x + 4)^2 + 3(3x + 4) - 3) - 1) - 1) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((-2(-1(-2cos(3\u03c0x - 2) + 1)^2 + 1(-2cos(3\u03c0x - 2) + 1) + 2) + 2)/(-3(-1(-2cos(3\u03c0x - 2) + 1)^2 + 1(-2cos(3\u03c0x - 2) + 1) + 2) + 4)) + 1| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(-1|(5(4x^2 + 2x + 5)^2 - 3(4x^2 + 2x + 5) - 3) + 2| - 2) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-5(3cos(2\u03c0(1|x - 2| + 1)) + 2)^2 - 4(3cos(2\u03c0(1|x - 2| + 1)) + 2) + 5) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((-2(-1sin(1\u03c0(-3|x - 3| + 3)) + 2) - 1)/(3(-1sin(1\u03c0(-3|x - 3| + 3)) + 2) + 1))^2 + 1((-2(-1sin(1\u03c0(-3|x - 3| + 3)) + 2) - 1)/(3(-1sin(1\u03c0(-3|x - 3| + 3)) + 2) + 1)) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(4(1sin(3\u03c0(1x - 1) - 3) - 3)^2 - 3(1sin(3\u03c0(1x - 1) - 3) - 3) + 3) + 1| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2(-2sin(3\u03c0(1x - 1) - 1)) + 5)^2 - 1(2(-2sin(3\u03c0(1x - 1) - 1)) + 5) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(2(1|(2x^2 + 1x) - 1| - 3)^2 - 5(1|(2x^2 + 1x) - 1| - 3)) + 1| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(2(-2|((-1x)/(3x + 1)) + 1| + 2) + 5) + 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4((1(2|(2x^2 + 3x + 1) - 1| - 2) - 1)/(-1(2|(2x^2 + 3x + 1) - 1| - 2) + 2))^2 + 1((1(2|(2x^2 + 3x + 1) - 1| - 2) - 1)/(-1(2|(2x^2 + 3x + 1) - 1| - 2) + 2)) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-4(3(-1|x + 2| + 1)^2 - 2(-1|x + 2| + 1) - 1)^2 - 2(3(-1|x + 2| + 1)^2 - 2(-1|x + 2| + 1) - 1) - 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-3(-3sin(2\u03c0(-2x - 3) + 2) + 2)^2 - 3))/(-3(-3(-3sin(2\u03c0(-2x - 3) + 2) + 2)^2 - 3) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-3(1|(1cos(3\u03c0x + 2) - 1) - 2| - 2)^2 + 5) - 1)/(-1(-3(1|(1cos(3\u03c0x + 2) - 1) - 2| - 2)^2 + 5) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1(-1(3cos(1\u03c0x + 1) + 3)^2 + 2(3cos(1\u03c0x + 1) + 3) - 4)^2 + 1(-1(3cos(1\u03c0x + 1) + 3)^2 + 2(3cos(1\u03c0x + 1) + 3) - 4) + 5) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1(-4(1sin(1\u03c0x + 2) - 1) + 3) + 4) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((-3(1(3cos(2\u03c0x - 1) + 1)^2 + 2(3cos(2\u03c0x - 1) + 1) - 1) + 1)/(1(1(3cos(2\u03c0x - 1) + 1)^2 + 2(3cos(2\u03c0x - 1) + 1) - 1) - 3))^2 - 5((-3(1(3cos(2\u03c0x - 1) + 1)^2 + 2(3cos(2\u03c0x - 1) + 1) - 1) + 1)/(1(1(3cos(2\u03c0x - 1) + 1)^2 + 2(3cos(2\u03c0x - 1) + 1) - 1) - 3)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3\u03c0(-3|(1|(-4x - 3) + 2| - 2) - 1| - 1) + 2) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2(2(-3sin(2\u03c0x) - 3) - 5) - 5)^2 + 3(-2(2(-3sin(2\u03c0x) - 3) - 5) - 5) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(5(1(-3cos(3\u03c0x) + 3)^2 + 2(-3cos(3\u03c0x) + 3) + 2) + 1) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(5(-2(2cos(3\u03c0x + 2) + 3) - 4) + 2) - 2)/(-3(5(-2(2cos(3\u03c0x + 2) + 3) - 4) + 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(4(-5(-2sin(2\u03c0x + 3) + 2) + 1) - 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|((-1(-2cos(2\u03c0(3x^2 - 5x)) + 3) - 2)/(-1(-2cos(2\u03c0(3x^2 - 5x)) + 3) - 1)) + 3| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-1|(2cos(3\u03c0((1x + 3)/(-3x))) - 1) + 3|)^2 - 2(-1|(2cos(3\u03c0((1x + 3)/(-3x))) - 1) + 3|) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(1|(-1|(-1sin(2\u03c0x - 3)) - 1|) + 2| + 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-2sin(1\u03c0(2|((1x + 3)/(-1x - 1)) - 1| + 3) - 3) + 1) + 1)/(-1(-2sin(1\u03c0(2|((1x + 3)/(-1x - 1)) - 1| + 3) - 3) + 1) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2\u03c0(-1|((-2((-1x)/(1x - 2)) - 2)/(2((-1x)/(1x - 2)) - 3)) - 2| + 2) + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(2((2((2x - 3)/(-2x - 1)) + 1)/(3((2x - 3)/(-2x - 1)))) - 2)) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(-5(-1(1x^2 - 5x + 4) + 5)^2 - 1(-1(1x^2 - 5x + 4) + 5) - 3) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-1|(1(4x^2 + 4x - 3)^2 - 4(4x^2 + 4x - 3) - 3) + 2| + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(3sin(1\u03c0(4(-2x - 1) - 1) + 1) + 2))/(-3(3sin(1\u03c0(4(-2x - 1) - 1) + 1) + 2) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((1(-2sin(2\u03c0(-5x^2 - 2x + 4) + 3) + 1) + 1)/(1(-2sin(2\u03c0(-5x^2 - 2x + 4) + 3) + 1) - 3)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(2|(3((-1x)/(3x + 1)) - 3) + 2| + 1)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-3cos(1\u03c0(-2|((2x + 2)/(-1x + 2)) + 3| + 2) - 1) + 1) - 1)/(1(-3cos(1\u03c0(-2|((2x + 2)/(-1x + 2)) + 3| + 2) - 1) + 1) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0(-4(5(3x + 2)^2 - 3(3x + 2) - 5) - 3) - 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-2(-2|(-3cos(1\u03c0x - 2) - 2) + 3| - 2) - 1) - 1)/(2(-2(-2|(-3cos(1\u03c0x - 2) - 2) + 3| - 2) - 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(3sin(2\u03c0(-1(3x - 3)^2 + 3(3x - 3) + 3) - 2) + 2) - 2)/(3(3sin(2\u03c0(-1(3x - 3)^2 + 3(3x - 3) + 3) - 2) + 2)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-2cos(2\u03c0((-3((-3x + 3)/(3x + 3)) + 1)/(-3((-3x + 3)/(3x + 3)) + 2)) + 2))^2 + 4(-2cos(2\u03c0((-3((-3x + 3)/(3x + 3)) + 1)/(-3((-3x + 3)/(3x + 3)) + 2)) + 2)) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-3|((3(1|x + 3| - 2) + 2)/(1(1|x + 3| - 2) - 3)) - 3|) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(-5(-3|(1x - 2) - 2| + 3) - 4) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3|(1|(-5x^2 - 5x - 1) + 3| - 2) + 1| - 1) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(4(-1|(-1x^2 - 5x - 5) - 1| + 2) + 4) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(1(3(-1x^2 + 1x + 4)^2 + 4(-1x^2 + 1x + 4) - 2) + 1) - 2) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3(-2(-3cos(1\u03c0x + 1) + 2)^2 - 1(-3cos(1\u03c0x + 1) + 2) + 5)^2 + 2(-2(-3cos(1\u03c0x + 1) + 2)^2 - 1(-3cos(1\u03c0x + 1) + 2) + 5) - 4) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-3cos(3\u03c0((-2(-1x + 4))/(-3(-1x + 4) - 3)) - 3) + 3)^2 + 5(-3cos(3\u03c0((-2(-1x + 4))/(-3(-1x + 4) - 3)) - 3) + 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-1((-2(3|x - 1|) + 1)/(-1(3|x - 1|) + 3))^2 + 4((-2(3|x - 1|) + 1)/(-1(3|x - 1|) + 3)) + 5) + 3| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(4(2sin(2\u03c0(1|x - 3| - 1) + 3) + 1)^2 - 5(2sin(2\u03c0(1|x - 3| - 1) + 3) + 1) + 5) + 2)/(3(4(2sin(2\u03c0(1|x - 3| - 1) + 3) + 1)^2 - 5(2sin(2\u03c0(1|x - 3| - 1) + 3) + 1) + 5) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2sin(2\u03c0(1|(-1|x - 2| - 1) + 3|) + 2) - 2)^2 + 1(2sin(2\u03c0(1|(-1|x - 2| - 1) + 3|) + 2) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(3cos(1\u03c0(1((3x - 3)/(-2x - 2)) - 1) + 2) + 2))/(1(3cos(1\u03c0(1((3x - 3)/(-2x - 2)) - 1) + 2) + 2) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-4(4(-3cos(1\u03c0x - 2) - 2) + 5)^2 + 4(4(-3cos(1\u03c0x - 2) - 2) + 5) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(2\u03c0(4(1(-2x - 5)^2 - 1(-2x - 5) - 5) + 3) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3(2sin(3\u03c0((-1x - 3)/(-1x + 2)) - 3) - 2) - 2)^2 - 4(-3(2sin(3\u03c0((-1x - 3)/(-1x + 2)) - 3) - 2) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((2(-2sin(2\u03c0(2x^2 - 3x - 3) - 3) - 2) + 2)/(1(-2sin(2\u03c0(2x^2 - 3x - 3) - 3) - 2) - 2)) - 2| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2((2(3cos(3\u03c0x) + 2) - 1)/(-3(3cos(3\u03c0x) + 2) - 2)) - 5) + 1| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(3(-3|(-1x^2 + 4x + 2) + 3| + 2) - 1) - 1) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3|(-2|(1|x - 1| + 2) - 2| + 2) + 2| - 3)^2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(4(2(1sin(2\u03c0x) - 1) + 2)^2 + 4(2(1sin(2\u03c0x) - 1) + 2) - 4)^2 + 5(4(2(1sin(2\u03c0x) - 1) + 2)^2 + 4(2(1sin(2\u03c0x) - 1) + 2) - 4) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(5(-1|(-1sin(2\u03c0x + 3) - 1) - 3| + 3)^2 - 4(-1|(-1sin(2\u03c0x + 3) - 1) - 3| + 3) - 4) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2cos(2\u03c0(1(-2x + 3)^2 - 5(-2x + 3) - 2) + 1) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1(-3(-2|x + 3| + 3)^2 + 5(-2|x + 3| + 3) + 1)^2 - 3(-3(-2|x + 3| + 3)^2 + 5(-2|x + 3| + 3) + 1) + 2) - 1)/(2(-1(-3(-2|x + 3| + 3)^2 + 5(-2|x + 3| + 3) + 1)^2 - 3(-3(-2|x + 3| + 3)^2 + 5(-2|x + 3| + 3) + 1) + 2) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2cos(3\u03c0(-2|(2|x| - 3) + 2| + 3) + 2) - 2)^2 + 5(-2cos(3\u03c0(-2|(2|x| - 3) + 2| + 3) + 2) - 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-3cos(1\u03c0(-3(-1|x - 2| - 3) - 5) + 1) + 3) - 1)/(-1(-3cos(1\u03c0(-3(-1|x - 2| - 3) - 5) + 1) + 3) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(2(-3(-1cos(3\u03c0x + 1) - 1) - 1) + 1)^2 + 4(2(-3(-1cos(3\u03c0x + 1) - 1) - 1) + 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((1(2|(-3|x - 3| + 2) - 1| - 1))/(2(2|(-3|x - 3| + 2) - 1| - 1) + 3))^2 + 1((1(2|(-3|x - 3| + 2) - 1| - 1))/(2(2|(-3|x - 3| + 2) - 1| - 1) + 3)) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0((-2((-1(2x - 4) - 2)/(-3(2x - 4))) + 2)/(-3((-1(2x - 4) - 2)/(-3(2x - 4))))) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-3sin(3\u03c0(-2(-1|x + 3| - 1)^2 + 2(-1|x + 3| - 1) - 1) + 3) - 1) + 2| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-1((3(-2cos(3\u03c0x + 2)) - 3)/(3(-2cos(3\u03c0x + 2)) + 1)) - 1)/(-2((3(-2cos(3\u03c0x + 2)) - 3)/(3(-2cos(3\u03c0x + 2)) + 1)) + 3)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-1(2(1cos(3\u03c0x + 1)))^2 - 1(2(1cos(3\u03c0x + 1)))) + 1| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2|((-2(-2sin(3\u03c0x - 2) + 2) + 3)/(-3(-2sin(3\u03c0x - 2) + 2) + 4)) - 2|) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2sin(1\u03c0(-1(-3x^2 + 2x + 5) - 2) - 2) + 3) + 3)/(2(-2sin(1\u03c0(-1(-3x^2 + 2x + 5) - 2) - 2) + 3) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((-3(3(-2cos(3\u03c0x + 1) - 2)^2 - 1(-2cos(3\u03c0x + 1) - 2) + 2) - 1)/(-2(3(-2cos(3\u03c0x + 1) - 2)^2 - 1(-2cos(3\u03c0x + 1) - 2) + 2) - 3)) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2cos(1\u03c0(3|((2x - 2)/(3x + 3)) - 3|) - 1) - 1) + 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(2(1sin(2\u03c0(-1|x - 3| + 1) + 3) + 2) - 2)^2 + 1(2(1sin(2\u03c0(-1|x - 3| + 1) + 3) + 2) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2|(-2(2cos(3\u03c0x)) - 2) - 2| - 2)^2 - 4(-2|(-2(2cos(3\u03c0x)) - 2) - 2| - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-1|(-1(-2x^2 - 2x + 2) - 2)|) + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3((-2(3sin(3\u03c0x - 1)) - 3)/(1(3sin(3\u03c0x - 1)) + 2)) - 3)^2 - 5(3((-2(3sin(3\u03c0x - 1)) - 3)/(1(3sin(3\u03c0x - 1)) + 2)) - 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0(1(1|(-1x^2 - 5x - 3) + 2| + 3)^2 - 2(1|(-1x^2 - 5x - 3) + 2| + 3) - 1) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-3(1|(2sin(1\u03c0x))| - 3) + 2) + 1| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(2\u03c0(-3|(-4((2x + 2)/(-2x + 1))^2 - 2((2x + 2)/(-2x + 1)) - 5) + 1| + 1) + 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2sin(2\u03c0(-3|(1x^2 + 3x - 2)| - 2) + 3) + 2) - 3)/(-1(-2sin(2\u03c0(-3|(1x^2 + 3x - 2)| - 2) + 3) + 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1|(3sin(2\u03c0((1x - 3)/(1x + 3)) - 1) - 3) - 3| + 3) + 3| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1(3(-2sin(2\u03c0x - 3) - 1)^2 + 2(-2sin(2\u03c0x - 3) - 1) + 1)^2 + 2(3(-2sin(2\u03c0x - 3) - 1)^2 + 2(-2sin(2\u03c0x - 3) - 1) + 1) + 2)^2 + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3((2(3sin(1\u03c0(-3|x| - 3) + 2)) + 1)/(2(3sin(1\u03c0(-3|x| - 3) + 2)) - 3)) + 2)/(3((2(3sin(1\u03c0(-3|x| - 3) + 2)) + 1)/(2(3sin(1\u03c0(-3|x| - 3) + 2)) - 3)) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0((3(-1(3|x + 3| - 2)^2 + 3(3|x + 3| - 2) - 3) + 3)/(-2(-1(3|x + 3| - 2)^2 + 3(3|x + 3| - 2) - 3))) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1|(-2sin(2\u03c0(-3|x + 1| + 2) - 1)) - 1| - 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(3\u03c0((-2(3x^2 + 4x - 3) + 3)/(-1(3x^2 + 4x - 3) - 1)) + 2) - 1) - 1)/(-2(-1sin(3\u03c0((-2(3x^2 + 4x - 3) + 3)/(-1(3x^2 + 4x - 3) - 1)) + 2) - 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-1cos(3\u03c0(-5(2x^2 - 4x - 1) - 2) - 1) + 2)^2 - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-5(2(-2cos(3\u03c0x) + 3) - 5)^2 + 1(2(-2cos(3\u03c0x) + 3) - 5) - 3) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(5(-1(1sin(2\u03c0x - 3) + 2) - 3) + 5) - 3| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1(2(1sin(3\u03c0x) + 2)^2 - 4(1sin(3\u03c0x) + 2) - 4) - 4) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2|(3|(-2sin(3\u03c0x + 3) - 1)| - 2) - 3| + 3) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2((-2((2(-1sin(3\u03c0x + 2) + 3))/(-2(-1sin(3\u03c0x + 2) + 3) - 3)) + 2)/(-2((2(-1sin(3\u03c0x + 2) + 3))/(-2(-1sin(3\u03c0x + 2) + 3) - 3)) - 1)) - 2)/(-1((-2((2(-1sin(3\u03c0x + 2) + 3))/(-2(-1sin(3\u03c0x + 2) + 3) - 3)) + 2)/(-2((2(-1sin(3\u03c0x + 2) + 3))/(-2(-1sin(3\u03c0x + 2) + 3) - 3)) - 1)) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3((2(1sin(3\u03c0(3x + 5)) + 2) + 3)/(-1(1sin(3\u03c0(3x + 5)) + 2) - 2)))/(3((2(1sin(3\u03c0(3x + 5)) + 2) + 3)/(-1(1sin(3\u03c0(3x + 5)) + 2) - 2)) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1((2(2cos(3\u03c0(2|x - 2| + 2) - 1) + 1) + 2)/(-2(2cos(3\u03c0(2|x - 2| + 2) - 1) + 1) + 1)) - 3)/(3((2(2cos(3\u03c0(2|x - 2| + 2) - 1) + 1) + 2)/(-2(2cos(3\u03c0(2|x - 2| + 2) - 1) + 1) + 1))) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-2sin(1\u03c0(-2|(5x^2 - 1x) + 1|) - 3) + 1) - 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3cos(1\u03c0(-5((3x + 1)/(-2x - 3)) - 2) + 2))^2 - 5(3cos(1\u03c0(-5((3x + 1)/(-2x - 3)) - 2) + 2)) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0(2|(-2|((-3x - 1)/(-2x - 2)) - 3| - 2) + 3| + 2) + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(5(-1sin(1\u03c0((-1x)/(-1x - 3)) + 2) - 2)^2 + 2(-1sin(1\u03c0((-1x)/(-1x - 3)) + 2) - 2) - 2)^2 + 4(5(-1sin(1\u03c0((-1x)/(-1x - 3)) + 2) - 2)^2 + 2(-1sin(1\u03c0((-1x)/(-1x - 3)) + 2) - 2) - 2) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1|(-2cos(3\u03c0((2x + 3)/(1x - 3))) + 1)| + 2)^2 + 3(1|(-2cos(3\u03c0((2x + 3)/(1x - 3))) + 1)| + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((3(-1(3sin(3\u03c0x + 2) - 2)^2 - 2(3sin(3\u03c0x + 2) - 2) - 1) + 3)/(-2(-1(3sin(3\u03c0x + 2) - 2)^2 - 2(3sin(3\u03c0x + 2) - 2) - 1) - 1)) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2sin(2\u03c0(2|(-3|x - 3| - 2) - 1| + 3) + 2) - 1)^2 + 3(2sin(2\u03c0(2|(-3|x - 3| - 2) - 1| + 3) + 2) - 1) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(1\u03c0((3((-3(-5x + 3))/(3(-5x + 3) - 1)) - 3)/(2((-3(-5x + 3))/(3(-5x + 3) - 1)) + 1)) - 1) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((-2(4(2cos(2\u03c0x + 1) - 3)) - 3)/(-3(4(2cos(2\u03c0x + 1) - 3)))) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|((-1(2sin(1\u03c0(3|x - 2| + 1) + 3) - 2) + 1)/(1(2sin(1\u03c0(3|x - 2| + 1) + 3) - 2) - 2)) - 3|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-5(3|(-2sin(2\u03c0x + 2)) - 2| - 1)^2 - 3(3|(-2sin(2\u03c0x + 2)) - 2| - 1) - 4) - 3| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(2\u03c0(3(2|(1x^2 - 2x + 3) - 1| - 2)^2 + 2(2|(1x^2 - 2x + 3) - 1| - 2) - 3)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-5(-3cos(3\u03c0((-1x + 3)/(-1x - 2)) + 3) - 3) + 1) + 1| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-5((1(-2sin(3\u03c0x) + 3))/(-3(-2sin(3\u03c0x) + 3) - 3)) - 5) + 2| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-3|((2(-2sin(2\u03c0x + 1) + 3) - 1)/(1(-2sin(2\u03c0x + 1) + 3) + 1)) + 3| + 1) + 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1((2(3|(1sin(2\u03c0x - 1) + 2) - 1| + 3) + 2)/(-1(3|(1sin(2\u03c0x - 1) + 2) - 1| + 3) + 2)) - 2)/(-3((2(3|(1sin(2\u03c0x - 1) + 2) - 1| + 3) + 2)/(-1(3|(1sin(2\u03c0x - 1) + 2) - 1| + 3) + 2)) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(3\u03c0(-5((3(-4x^2 + 1x - 1))/(2(-4x^2 + 1x - 1) + 1)) - 1) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(1(-1sin(3\u03c0(2x + 5) + 3) - 2) - 5) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0((-1((2(1|x| - 1) + 1)/(-3(1|x| - 1) + 2)) + 3)/(3((2(1|x| - 1) + 1)/(-3(1|x| - 1) + 2)) - 2)) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(3(-3cos(3\u03c0(-2|x - 3|)) + 3)^2 + 2(-3cos(3\u03c0(-2|x - 3|)) + 3))^2 + 2(3(-3cos(3\u03c0(-2|x - 3|)) + 3)^2 + 2(-3cos(3\u03c0(-2|x - 3|)) + 3)) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2sin(2\u03c0(-3|((-3x + 3)/(-2x + 1)) + 2| - 2) - 1) + 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2|(-1(2cos(3\u03c0x - 3) + 3)^2 - 1(2cos(3\u03c0x - 3) + 3) + 4)| - 2)^2 - 5(-2|(-1(2cos(3\u03c0x - 3) + 3)^2 - 1(2cos(3\u03c0x - 3) + 3) + 4)| - 2) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3(-4(3cos(2\u03c0x - 2) + 2)^2 + 1(3cos(2\u03c0x - 2) + 2) + 2)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(1\u03c0(-5(1|(-2x - 4) + 1| - 3) - 1) - 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1sin(2\u03c0(2|((1x)/(-1x - 2)) - 3|) + 3) - 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3((2(-1cos(3\u03c0(2x^2 + 4x + 5) - 3) + 2) + 1)/(2(-1cos(3\u03c0(2x^2 + 4x + 5) - 3) + 2) + 3)) - 1)/(-1((2(-1cos(3\u03c0(2x^2 + 4x + 5) - 3) + 2) + 1)/(2(-1cos(3\u03c0(2x^2 + 4x + 5) - 3) + 2) + 3)) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-2sin(1\u03c0(-1|((-2x - 1)/(1x - 2)) - 3| - 1) - 3) - 1) + 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2sin(1\u03c0((-2((1x - 3)/(-2x - 3)) - 3)/(1((1x - 3)/(-2x - 3)) + 3)) - 3)) + 1)/(1(-2sin(1\u03c0((-2((1x - 3)/(-2x - 3)) - 3)/(1((1x - 3)/(-2x - 3)) + 3)) - 3))), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1cos(1\u03c0(1|((-3x - 1)/(-2x - 3)) + 2|) + 3) - 3)^2 + 1(-1cos(1\u03c0(1|((-3x - 1)/(-2x - 3)) + 2|) + 3) - 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2cos(1\u03c0(5(2|x + 2|) + 2) - 2))^2 - 5(2cos(1\u03c0(5(2|x + 2|) + 2) - 2)) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(5(3sin(3\u03c0(1|x - 3| - 2) - 2) - 3)^2 - 1) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2sin(1\u03c0((2(1x) - 2)/(-2(1x))) - 3) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2|(3sin(1\u03c0(1|x - 2| + 3) + 3) - 2) - 3| + 1)^2 - 4(2|(3sin(1\u03c0(1|x - 2| + 3) + 3) - 2) - 3| + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2|(-4(-1cos(1\u03c0x - 1) + 1)^2 + 2(-1cos(1\u03c0x - 1) + 1) - 2) - 3| + 1) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2(2sin(3\u03c0((2x + 2)/(-1x + 3))) + 2) - 2)^2 + 3(2(2sin(3\u03c0((2x + 2)/(-1x + 3))) + 2) - 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-3|(-2(-1|x - 1| - 1)^2 - 2(-1|x - 1| - 1) + 4)| + 1) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(5(1cos(1\u03c0((-3x - 3)/(-3x + 4)) + 1) + 1) + 4) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(2(-2sin(2\u03c0((2x - 2)/(-3x - 2)) + 2) - 3)))/(1(2(-2sin(2\u03c0((2x - 2)/(-3x - 2)) + 2) - 3)) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(4(2|((3x + 1)/(-1x + 2))| + 1) - 2) - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2|(4(-1|x - 2| + 2)^2 - 4(-1|x - 2| + 2)) - 3| - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(1\u03c0(-3(4((-3x + 2)/(2x + 3)) + 2)^2 - 1(4((-3x + 2)/(2x + 3)) + 2) + 2) - 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(-4(3|((2x - 2)/(-1x)) - 2| - 2) + 3) - 1) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(4(-1cos(3\u03c0((2x + 1)/(1x + 3)) - 1) + 2) + 4) + 3)/(-1(4(-1cos(3\u03c0((2x + 1)/(1x + 3)) - 1) + 2) + 4) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2|(-3|(-3|x - 2| + 2) + 3| + 1) + 1| - 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(2(-1cos(1\u03c0((3x - 1)/(2x - 3))) - 2) - 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3|(-1(2cos(3\u03c0x + 3) + 2) - 4) + 2|)^2 - 2(-3|(-1(2cos(3\u03c0x + 3) + 2) - 4) + 2|) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-2cos(1\u03c0(1(-3|x - 1| + 1) - 3) + 3)) - 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1|(-2sin(3\u03c0(3x - 5)) - 1) + 1| + 2) - 2| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1cos(2\u03c0(-2|((-3x + 2)/(-1x - 3)) + 3| - 2) - 1) + 2)^2 + 4(-1cos(2\u03c0(-2|((-3x + 2)/(-1x - 3)) + 3| - 2) - 1) + 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-3|(1cos(2\u03c0((-1x - 3)/(3x + 3))) + 3) + 3| + 3) - 1)/(-2(-3|(1cos(2\u03c0((-1x - 3)/(3x + 3))) + 3) + 3| + 3) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(2(3|(3sin(2\u03c0x - 3) + 1) + 3| - 2)) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2sin(2\u03c0(-3((-1x - 2)/(2x + 3))) - 3) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0((-3((2(5x^2 + 5x) - 2)/(1(5x^2 + 5x) + 1)) - 1)/(-3((2(5x^2 + 5x) - 2)/(1(5x^2 + 5x) + 1)) + 4)) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-2|(-1(-5x^2 - 1x + 1)^2 - 5(-5x^2 - 1x + 1) - 5)| + 1) - 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0(-1|(-2|(-5x + 2) - 3| - 1) - 3| - 1) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1|(-2(5x^2 + 1x - 5) - 3) + 3| + 2) + 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1(-3cos(2\u03c0(3|x - 2| - 1) + 2)) - 4) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(2|(-3(-2|x - 2| + 3) + 3) - 1| - 2) + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3cos(1\u03c0(-2|(3x^2 + 5x + 5) - 3| - 2) + 1) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(1|((-2(2cos(2\u03c0x - 2)) + 1)/(1(2cos(2\u03c0x - 2)) - 3)) + 3| - 1) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(2\u03c0(3((-1(-3|x + 1| - 3))/(-1(-3|x + 1| - 3) - 3)) + 2) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(2(-2sin(1\u03c0(-1|x + 1| + 3) - 1))^2 - 3(-2sin(1\u03c0(-1|x + 1| + 3) - 1)) - 4) - 3| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-5(1cos(1\u03c0(1x + 5) + 2) - 2) - 2) + 3| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3|((2(3sin(2\u03c0x + 3) - 3) - 1)/(-1(3sin(2\u03c0x + 3) - 3) - 1)) - 1|) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0(-1(1(5x - 5))^2 + 3(1(5x - 5)) + 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2\u03c0(1((2(2|x - 1|) - 3)/(2(2|x - 1|) - 1))^2 + 1((2(2|x - 1|) - 3)/(2(2|x - 1|) - 1)) + 1) - 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1\u03c0(-4((3(1x + 4) + 1)/(3(1x + 4) - 1))^2 + 5((3(1x + 4) + 1)/(3(1x + 4) - 1)) - 1) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(2cos(1\u03c0(4((3x - 2)/(-3x - 3)) - 5) + 2) + 1)^2 - 2(2cos(1\u03c0(4((3x - 2)/(-3x - 3)) - 5) + 2) + 1) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((1(2sin(1\u03c0((2x - 1)/(-1x - 1))) - 2))/(-2(2sin(1\u03c0((2x - 1)/(-1x - 1))) - 2) - 1)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(2|(-1(2|x + 3| - 2) - 2) + 2| + 1) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(1|(3cos(1\u03c0(-2x^2 + 4x - 4) + 1) + 3) - 3| - 3)| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1cos(2\u03c0((2(2|x + 3|) - 2)/(-2(2|x + 3|))) - 2)) - 3)/(3(-1cos(2\u03c0((2(2|x + 3|) - 2)/(-2(2|x + 3|))) - 2)) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(1cos(1\u03c0(-1((2x - 3)/(-1x - 3)) - 3) - 3) - 2)^2 + 2(1cos(1\u03c0(-1((2x - 3)/(-1x - 3)) - 3) - 3) - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(5(1|(2cos(3\u03c0x - 1) + 2) + 2|)^2 - 5(1|(2cos(3\u03c0x - 1) + 2) + 2|) + 5) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(4(2cos(2\u03c0(-1|x - 2| - 1) + 1) + 3) - 3) + 2)/(2(4(2cos(2\u03c0(-1|x - 2| - 1) + 1) + 3) - 3) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(3sin(1\u03c0(2|(5x^2 + 5x - 2) + 3| - 1) + 1) + 1) - 3)/(-2(3sin(1\u03c0(2|(5x^2 + 5x - 2) + 3| - 1) + 1) + 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2(1sin(2\u03c0(-5x^2 - 4x - 3) + 1) - 3) - 4) + 3)/(2(-2(1sin(2\u03c0(-5x^2 - 4x - 3) + 1) - 3) - 4) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(1|(-4(2|x - 1| + 2) + 2) - 3| - 2) - 1) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(1(-1(-3cos(2\u03c0x - 3) - 1)^2 + 2(-3cos(2\u03c0x - 3) - 1) - 1)^2 + 4(-1(-3cos(2\u03c0x - 3) - 1)^2 + 2(-3cos(2\u03c0x - 3) - 1) - 1) - 4))/(2(1(-1(-3cos(2\u03c0x - 3) - 1)^2 + 2(-3cos(2\u03c0x - 3) - 1) - 1)^2 + 4(-1(-3cos(2\u03c0x - 3) - 1)^2 + 2(-3cos(2\u03c0x - 3) - 1) - 1) - 4) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(2|(-3cos(1\u03c0(-3|x - 3| - 1) + 1) - 2) - 2| + 2) + 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-2sin(1\u03c0(1|((-2x - 3)/(-1x + 3)) - 3| + 2) - 1) + 2) + 1)/(-2(-2sin(1\u03c0(1|((-2x - 3)/(-1x + 3)) - 3| + 2) - 1) + 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(1(2sin(3\u03c0(-4x - 4) - 3) - 3) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-3sin(2\u03c0(-4(-1x^2 - 4x - 3)^2 - 2(-1x^2 - 4x - 3) - 1) - 2) + 3) + 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3sin(1\u03c0(1|((-1x - 3)/(-3x + 2)) - 3|) + 2) - 3)^2 - 5(3sin(1\u03c0(1|((-1x - 3)/(-3x + 2)) - 3|) + 2) - 3) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-4(1cos(1\u03c0(-2|x - 1| + 2) + 1) - 3) - 4), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-2cos(1\u03c0(-3|(-1x - 5) + 1| + 2) - 1)) - 2)/(-1(-2cos(1\u03c0(-3|(-1x - 5) + 1| + 2) - 1)) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1(-3cos(1\u03c0((-3x - 3)/(2x + 1)) - 1)) - 2)^2 + 3(-1(-3cos(1\u03c0((-3x - 3)/(2x + 1)) - 1)) - 2) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1\u03c0((-3((2((-2x - 1)/(-3x - 2)) - 2)/(-3((-2x - 1)/(-3x - 2)))) - 3)/(-1((2((-2x - 1)/(-3x - 2)) - 2)/(-3((-2x - 1)/(-3x - 2)))) + 3))) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1|(-3cos(2\u03c0((1x + 2)/(2x - 3)) + 1) - 2) + 1| + 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(5(-2|((-2x)/(-2x + 3))| - 1) - 1) - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(1|(-3|(3cos(3\u03c0x + 3) + 2) + 3| + 2) - 3| - 2) + 2| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-4(3sin(1\u03c0((-2x - 3)/(1x)) - 1) + 1) - 3) + 2|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-2|(2|(-2|x + 2| + 3) - 3| + 2) + 3| + 1) - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(3\u03c0(-2(-1|(4x^2 - 2x + 4) + 1| - 2) - 5) + 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(1(5((2x + 2)/(-3x - 1)) + 3)^2 - 5(5((2x + 2)/(-3x - 1)) + 3) + 2)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0((-1(5(-1x^2 - 4x)^2 - 4(-1x^2 - 4x) - 1) + 3)/(1(5(-1x^2 - 4x)^2 - 4(-1x^2 - 4x) - 1)))) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-2(2(1sin(3\u03c0x - 2)) - 4) - 1)/(1(2(1sin(3\u03c0x - 2)) - 4) - 2)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-3(2|(-3sin(2\u03c0x + 2) - 3) - 1| + 1) - 1)/(-1(2|(-3sin(2\u03c0x + 2) - 3) - 1| + 1) + 2)) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1|(-3(-3cos(3\u03c0x - 2) - 3)^2 + 5(-3cos(3\u03c0x - 2) - 3) + 1)| + 3) + 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(4((-3(1sin(3\u03c0x + 3) - 3))/(-1(1sin(3\u03c0x + 3) - 3) - 3))^2 - 4((-3(1sin(3\u03c0x + 3) - 3))/(-1(1sin(3\u03c0x + 3) - 3) - 3)) + 4) + 1)/(3(4((-3(1sin(3\u03c0x + 3) - 3))/(-1(1sin(3\u03c0x + 3) - 3) - 3))^2 - 4((-3(1sin(3\u03c0x + 3) - 3))/(-1(1sin(3\u03c0x + 3) - 3) - 3)) + 4)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3((-2(-2sin(3\u03c0(2x^2 - 1x + 1) + 3) - 1) + 2)/(2(-2sin(3\u03c0(2x^2 - 1x + 1) + 3) - 1) - 1)) - 1)/(3((-2(-2sin(3\u03c0(2x^2 - 1x + 1) + 3) - 1) + 2)/(2(-2sin(3\u03c0(2x^2 - 1x + 1) + 3) - 1) - 1))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(1|(2sin(3\u03c0(5x^2 + 1x + 5) + 1) - 1) - 2|)^2 + 5(1|(2sin(3\u03c0(5x^2 + 1x + 5) + 1) - 1) - 2|) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1|(-2cos(2\u03c0((1x - 3)/(2x - 1)) + 3) - 1) - 1| - 1) + 3| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2(1(1x^2 + 2x + 1) - 4)^2 + 3(1(1x^2 + 2x + 1) - 4) + 3)^2 + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(1(2(1cos(1\u03c0x + 1))^2 + 2(1cos(1\u03c0x + 1)) - 5)^2 + 1(2(1cos(1\u03c0x + 1))^2 + 2(1cos(1\u03c0x + 1)) - 5) + 2) + 1)/(-2(1(2(1cos(1\u03c0x + 1))^2 + 2(1cos(1\u03c0x + 1)) - 5)^2 + 1(2(1cos(1\u03c0x + 1))^2 + 2(1cos(1\u03c0x + 1)) - 5) + 2) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1(3sin(1\u03c0(-3|x + 2| + 1)) + 3)^2 - 3(3sin(1\u03c0(-3|x + 2| + 1)) + 3))^2 + 2(1(3sin(1\u03c0(-3|x + 2| + 1)) + 3)^2 - 3(3sin(1\u03c0(-3|x + 2| + 1)) + 3)) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4((2(2sin(3\u03c0((2x + 1)/(2x - 1)) - 2) - 1) - 3)/(-3(2sin(3\u03c0((2x + 1)/(2x - 1)) - 2) - 1) - 3))^2 + 5((2(2sin(3\u03c0((2x + 1)/(2x - 1)) - 2) - 1) - 3)/(-3(2sin(3\u03c0((2x + 1)/(2x - 1)) - 2) - 1) - 3)) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((2(3|(2sin(2\u03c0x))| - 3))/(3(3|(2sin(2\u03c0x))| - 3) - 2)) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(2(1|((1x + 1)/(3x)) + 2| + 1) - 1) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2|((-1(-1sin(2\u03c0x - 1) - 3) + 1)/(1(-1sin(2\u03c0x - 1) - 3) + 3)) + 2| + 2) + 3| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2(-1cos(3\u03c0(-3x + 2)) - 3)^2 + 5(-1cos(3\u03c0(-3x + 2)) - 3) - 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((-3(2(-2sin(2\u03c0x + 3) - 1)^2 - 5(-2sin(2\u03c0x + 3) - 1)) - 1)/(3(2(-2sin(2\u03c0x + 3) - 1)^2 - 5(-2sin(2\u03c0x + 3) - 1)) + 3))^2 + 3((-3(2(-2sin(2\u03c0x + 3) - 1)^2 - 5(-2sin(2\u03c0x + 3) - 1)) - 1)/(3(2(-2sin(2\u03c0x + 3) - 1)^2 - 5(-2sin(2\u03c0x + 3) - 1)) + 3)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(3\u03c0(-1|(-2|((-3x + 3)/(1x - 2)) - 2| + 1) + 3| + 3) - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2sin(2\u03c0(-2((-2x - 1)/(-1x - 3))^2 - 3((-2x - 1)/(-1x - 3)) + 3)) - 2)^2 + 4(-2sin(2\u03c0(-2((-2x - 1)/(-1x - 3))^2 - 3((-2x - 1)/(-1x - 3)) + 3)) - 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-2|(-2cos(2\u03c0(5x - 5) - 3) - 1) - 2| - 3) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(2sin(2\u03c0(-2|(3x + 4) + 1| - 1) + 3) + 3)^2 + 2(2sin(2\u03c0(-2|(3x + 4) + 1| - 1) + 3) + 3) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-1cos(3\u03c0(5((3x + 3)/(-2x + 1))^2 + 3((3x + 3)/(-2x + 1)) - 4) - 2) - 3) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((-3(1cos(2\u03c0((-3x + 3)/(-2x)) - 2) - 1) + 3)/(-1(1cos(2\u03c0((-3x + 3)/(-2x)) - 2) - 1) + 3))| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(1\u03c0(-2|(3|(-2x^2 - 5x - 2) - 2| + 3) + 1| - 3)) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0(-1(-5(3|x - 2| - 3) - 2)^2 + 1) + 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2cos(2\u03c0(-2|((3x + 2)/(1x - 3)) + 1| - 2) - 2) + 1) + 1)/(-2(2cos(2\u03c0(-2|((3x + 2)/(1x - 3)) + 1| - 2) - 2) + 1) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|((2(1|(2sin(2\u03c0x - 3) - 2) - 3|) - 2)/(1(1|(2sin(2\u03c0x - 3) - 2) - 3|) + 3)) - 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0((-3(-1(-1x^2 + 3x - 4) - 1) - 3)/(-3(-1(-1x^2 + 3x - 4) - 1) + 2)) - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((2(-1(-2sin(2\u03c0x - 2) + 2)^2 - 1(-2sin(2\u03c0x - 2) + 2) + 4))/(1(-1(-2sin(2\u03c0x - 2) + 2)^2 - 1(-2sin(2\u03c0x - 2) + 2) + 4) + 3)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((1(-2sin(3\u03c0(-3|x| + 2) - 3) - 1) - 1)/(1(-2sin(3\u03c0(-3|x| + 2) - 3) - 1))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-3sin(3\u03c0(-3(-2|x + 3| + 3)^2 - 3(-2|x + 3| + 3) + 5))) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(1(-1cos(2\u03c0(3|x| - 2) - 3) - 3) + 5) - 1)/(2(1(-1cos(2\u03c0(3|x| - 2) - 3) - 3) + 5) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(4(3(-3cos(3\u03c0x - 2)) - 1)) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-3(1cos(2\u03c0(5x - 4)) + 2) - 4) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "108", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(2sin(1\u03c0(1(2|x - 1| - 1)^2 - 5(2|x - 1| - 1) - 3) - 2) - 1) + 3)/(1(2sin(1\u03c0(1(2|x - 1| - 1)^2 - 5(2|x - 1| - 1) - 3) - 2) - 1) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0((3(3((3x - 3)/(-1x - 2))^2 + 4((3x - 3)/(-1x - 2)) + 5))/(-1(3((3x - 3)/(-1x - 2))^2 + 4((3x - 3)/(-1x - 2)) + 5) + 2)) - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1|(-1sin(1\u03c0(-4x^2 - 3x + 2) - 1) - 3) + 2| + 1) + 1)/(2(1|(-1sin(1\u03c0(-4x^2 - 3x + 2) - 1) - 3) + 2| + 1) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-4(2|(-2|x - 3| - 1) + 2| - 1)^2 + 5(2|(-2|x - 3| - 1) + 2| - 1) - 1) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1(3sin(3\u03c0((3x + 1)/(2x)) + 2) - 3) + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-1(5(3sin(3\u03c0x + 3) - 1)^2 - 2(3sin(3\u03c0x + 3) - 1) - 2)^2 - 3(5(3sin(3\u03c0x + 3) - 1)^2 - 2(3sin(3\u03c0x + 3) - 1) - 2) - 3) - 2)/(1(-1(5(3sin(3\u03c0x + 3) - 1)^2 - 2(3sin(3\u03c0x + 3) - 1) - 2)^2 - 3(5(3sin(3\u03c0x + 3) - 1)^2 - 2(3sin(3\u03c0x + 3) - 1) - 2) - 3) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2(-2(-2sin(1\u03c0x + 2)) + 3)^2 + 4(-2(-2sin(1\u03c0x + 2)) + 3) + 1)^2 + 2(-2(-2(-2sin(1\u03c0x + 2)) + 3)^2 + 4(-2(-2sin(1\u03c0x + 2)) + 3) + 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3|(5(1|x - 2| - 1)) - 1| - 2)^2 - 4(3|(5(1|x - 2| - 1)) - 1| - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-3sin(2\u03c0(1((2x)/(1x - 3))^2 - 3((2x)/(1x - 3)))) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2sin(3\u03c0((-2(-3|x - 1|) + 2)/(-1(-3|x - 1|) - 3)) - 2) + 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3|((-2(2cos(2\u03c0x)) - 1)/(2(2cos(2\u03c0x)) + 3)) + 2|)^2 - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((2(2(1sin(1\u03c0x - 1) - 3)^2 + 2(1sin(1\u03c0x - 1) - 3) - 4) + 2)/(2(2(1sin(1\u03c0x - 1) - 3)^2 + 2(1sin(1\u03c0x - 1) - 3) - 4) - 1)) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4((-1(2(2cos(2\u03c0x + 1) + 3)^2 + 5(2cos(2\u03c0x + 1) + 3) - 1) + 3)/(3(2(2cos(2\u03c0x + 1) + 3)^2 + 5(2cos(2\u03c0x + 1) + 3) - 1) + 2)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(3sin(1\u03c0(-1|(1x - 1) + 2| + 1) + 1) - 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(5(-2sin(2\u03c0(1x + 3) - 3) - 3) - 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(1(-2sin(1\u03c0(2x - 1) - 3) - 2) + 3) - 1| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(3sin(2\u03c0(-1|(4x^2 - 2x - 2) + 2| + 1) + 1) + 1) - 1)/(2(3sin(2\u03c0(-1|(4x^2 - 2x - 2) + 2| + 1) + 1) + 1) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1|((3(-2cos(1\u03c0x - 2) - 1))/(1(-2cos(1\u03c0x - 2) - 1) + 2))| - 2)^2 + 2(1|((3(-2cos(1\u03c0x - 2) - 1))/(1(-2cos(1\u03c0x - 2) - 1) + 2))| - 2) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-2cos(3\u03c0((-3(-5x^2 + 5) - 1)/(-1(-5x^2 + 5) + 3))) + 3) - 1| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4((3(4(-1cos(2\u03c0x) - 3) + 3) + 2)/(-1(4(-1cos(2\u03c0x) - 3) + 3) + 2))^2 + 4((3(4(-1cos(2\u03c0x) - 3) + 3) + 2)/(-1(4(-1cos(2\u03c0x) - 3) + 3) + 2)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(1|(-4x^2 + 5x + 4) + 1| - 1) + 4)^2 - 4(-1(1|(-4x^2 + 5x + 4) + 1| - 1) + 4) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1|((-1(1cos(2\u03c0x - 1) - 3) + 2)/(1(1cos(2\u03c0x - 1) - 3) - 3)) + 2| - 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3((3(-2sin(3\u03c0x + 1) + 3) + 2)/(3(-2sin(3\u03c0x + 1) + 3)))^2 - 1((3(-2sin(3\u03c0x + 1) + 3) + 2)/(3(-2sin(3\u03c0x + 1) + 3))) + 5) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(3((-2(2sin(3\u03c0x - 2) + 3))/(-1(2sin(3\u03c0x - 2) + 3) - 2)) + 1) + 2)/(-2(3((-2(2sin(3\u03c0x - 2) + 3))/(-1(2sin(3\u03c0x - 2) + 3) - 2)) + 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(1|(3((-2x - 1)/(2x + 3)) + 5) - 2| + 1) - 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5((-3(-2|(2sin(2\u03c0x) - 3) - 2| - 1) + 2)/(-1(-2|(2sin(2\u03c0x) - 3) - 2| - 1))) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-4(-1cos(2\u03c0(-3x - 1) + 3) + 1) + 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(3cos(3\u03c0(3(-1|x - 2| + 3) + 3))) - 2)/(3(3cos(3\u03c0(3(-1|x - 2| + 3) + 3))) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(1(-3sin(2\u03c0(5x^2 - 3x + 3) - 3) - 3)^2 - 1(-3sin(2\u03c0(5x^2 - 3x + 3) - 3) - 3)))/(-1(1(-3sin(2\u03c0(5x^2 - 3x + 3) - 3) - 3)^2 - 1(-3sin(2\u03c0(5x^2 - 3x + 3) - 3) - 3)) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(2sin(1\u03c0(1(-3|x + 2|)^2 + 4(-3|x + 2|) + 3) + 3) + 3) + 3)/(3(2sin(1\u03c0(1(-3|x + 2|)^2 + 4(-3|x + 2|) + 3) + 3) + 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((-2(3(-1cos(2\u03c0x + 2) + 2) + 5) + 1)/(3(3(-1cos(2\u03c0x + 2) + 2) + 5)))^2 - 1((-2(3(-1cos(2\u03c0x + 2) + 2) + 5) + 1)/(3(3(-1cos(2\u03c0x + 2) + 2) + 5))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2(2sin(1\u03c0(-1x^2 + 4x + 3) - 3) + 1) + 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1(1sin(2\u03c0(-4x - 2) - 3) + 3) - 4) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-5((1(-2cos(3\u03c0x + 3)) + 2)/(3(-2cos(3\u03c0x + 3)) + 1)) + 5) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(1|(-3sin(2\u03c0(3x - 5)) - 1) - 3| - 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(4((-1(-2sin(1\u03c0x - 3) - 1) + 3)/(2(-2sin(1\u03c0x - 3) - 1) - 1)) + 2) - 3)/(-1(4((-1(-2sin(1\u03c0x - 3) - 1) + 3)/(2(-2sin(1\u03c0x - 3) - 1) - 1)) + 2)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(-3(3(3|x| - 3) + 1)^2 - 2(3(3|x| - 3) + 1) + 1) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-3(1cos(2\u03c0(-5x + 4) + 2) - 1)^2 + 1(1cos(2\u03c0(-5x + 4) + 2) - 1) + 3)^2 + 1(-3(1cos(2\u03c0(-5x + 4) + 2) - 1)^2 + 1(1cos(2\u03c0(-5x + 4) + 2) - 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(-3|(-1|(-3|x + 1| - 3)| - 2) - 1| + 2) + 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1cos(1\u03c0(-1|(-2|x + 2| + 1) - 3| - 1) - 3) - 1) - 2| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-2|(-3cos(3\u03c0((1x)/(-1x - 3))) + 2) - 1| + 1) + 2)/(-2(-2|(-3cos(3\u03c0((1x)/(-1x - 3))) + 2) - 1| + 1) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(1\u03c0(-2|((2(-2|x + 3| + 1) - 2)/(-2(-2|x + 3| + 1) - 1)) + 3| + 3) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-2(-2|(-2sin(2\u03c0x) + 3) + 2| - 3) - 1)/(-3(-2|(-2sin(2\u03c0x) + 3) + 2| - 3) + 1)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0(-1|(4(-4x + 4) - 1)| + 2) - 1) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2sin(2\u03c0(-4(3|x + 3| - 3)^2 - 1(3|x + 3| - 3) - 3) - 3) - 2)^2 + 3(2sin(2\u03c0(-4(3|x + 3| - 3)^2 - 1(3|x + 3| - 3) - 3) - 3) - 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(1|((1(1sin(2\u03c0x - 1) + 3) - 3)/(-3(1sin(2\u03c0x - 1) + 3) - 3)) + 3| + 2) - 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(3|((-1((1x + 3)/(-2x + 3)) - 3)/(-3((1x + 3)/(-2x + 3)))) + 1| - 2) - 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3sin(2\u03c0(-1|(5x^2 - 1x + 5)| - 1) - 2) - 2)^2 - 2(3sin(2\u03c0(-1|(5x^2 - 1x + 5)| - 1) - 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-5(1cos(2\u03c0((-1x - 3)/(-2x - 3)) + 3) - 1) + 5) + 3)/(-1(-5(1cos(2\u03c0((-1x - 3)/(-2x - 3)) + 3) - 1) + 5) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1cos(3\u03c0(3|(-1x - 3) + 3| + 2) - 2) - 3) + 1| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-3(-3sin(1\u03c0(-3|x - 2| + 2) + 2) - 2) - 3)/(-3(-3sin(1\u03c0(-3|x - 2| + 2) + 2) - 2)))^2 + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-5(-4(-2|x - 1| + 3)^2 - 2(-2|x - 1| + 3) + 2)^2 + 3(-4(-2|x - 1| + 3)^2 - 2(-2|x - 1| + 3) + 2)))/(1(-5(-4(-2|x - 1| + 3)^2 - 2(-2|x - 1| + 3) + 2)^2 + 3(-4(-2|x - 1| + 3)^2 - 2(-2|x - 1| + 3) + 2)) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2sin(1\u03c0(-3(3|x - 2| + 1)^2 - 3(3|x - 2| + 1) + 1) + 1) - 3) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0(-1|(3(-4x + 2) - 1) - 2| + 1) + 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-2((-2(2sin(2\u03c0x + 2) + 3) + 1)/(-3(2sin(2\u03c0x + 2) + 3) - 3)) - 2)^2 - 5(-2((-2(2sin(2\u03c0x + 2) + 3) + 1)/(-3(2sin(2\u03c0x + 2) + 3) - 3)) - 2) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((-2(-1sin(3\u03c0(-3|x - 1| - 2) - 3) + 3) - 1)/(2(-1sin(3\u03c0(-3|x - 1| - 2) - 3) + 3) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "92", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(1|((-3((1x + 2)/(2x - 1)))/(-3((1x + 2)/(2x - 1)) + 4)) - 1| - 2) - 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(5(-1((-1x + 3)/(2x - 3))^2 - 5((-1x + 3)/(2x - 3)))^2 + 3(-1((-1x + 3)/(2x - 3))^2 - 5((-1x + 3)/(2x - 3))) - 5) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-1cos(1\u03c0(1((-3x + 1)/(2x - 3))^2 + 4((-3x + 1)/(2x - 3)) - 5) - 3) - 1))/(3(-1cos(1\u03c0(1((-3x + 1)/(2x - 3))^2 + 4((-3x + 1)/(2x - 3)) - 5) - 3) - 1) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1((1(1(-3sin(1\u03c0x - 1) - 1) - 4) - 1)/(-1(1(-3sin(1\u03c0x - 1) - 1) - 4))))/(-1((1(1(-3sin(1\u03c0x - 1) - 1) - 4) - 1)/(-1(1(-3sin(1\u03c0x - 1) - 1) - 4))) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(1cos(2\u03c0(-3((3x)/(1x + 3)) - 1) - 3) + 1) + 3| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2(-5(-3cos(1\u03c0x + 2) - 2) - 2)^2 + 4(-5(-3cos(1\u03c0x + 2) - 2) - 2) - 5) + 1)/(2(-2(-5(-3cos(1\u03c0x + 2) - 2) - 2)^2 + 4(-5(-3cos(1\u03c0x + 2) - 2) - 2) - 5) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3cos(3\u03c0(-1((2x + 1)/(2x + 3))^2 - 2((2x + 1)/(2x + 3)) - 5) - 2) + 3)^2 + 2(3cos(3\u03c0(-1((2x + 1)/(2x + 3))^2 - 2((2x + 1)/(2x + 3)) - 5) - 2) + 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1|(-1|(1sin(1\u03c0x - 1) + 1) - 3| - 1) + 3| - 3) - 2)/(-1(-1|(-1|(1sin(1\u03c0x - 1) + 1) - 3| - 1) + 3| - 3) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3\u03c0((3((-1(3x - 1))/(-1(3x - 1) - 3)) - 2)/(-2((-1(3x - 1))/(-1(3x - 1) - 3)) + 3)) + 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(5(-1sin(3\u03c0(-1x^2 - 2x + 1)) - 2)^2 - 2(-1sin(3\u03c0(-1x^2 - 2x + 1)) - 2) + 4) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-3(-1|(-2sin(3\u03c0x - 1) - 1)|) + 1)/(2(-1|(-2sin(3\u03c0x - 1) - 1)|))) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((1(3|(1|x - 2| - 2) - 3| - 2) + 3)/(-2(3|(1|x - 2| - 2) - 3| - 2) - 1))^2 - 4((1(3|(1|x - 2| - 2) - 3| - 2) + 3)/(-2(3|(1|x - 2| - 2) - 3| - 2) - 1)) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2sin(2\u03c0((1(-3x))/(1(-3x) - 2)) + 3))^2 - 5(-2sin(2\u03c0((1(-3x))/(1(-3x) - 2)) + 3)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-5(3(2cos(2\u03c0x + 1) + 2) + 3) - 4) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(1(3cos(2\u03c0((2x - 2)/(1x)) + 1) + 3)^2 - 2(3cos(2\u03c0((2x - 2)/(1x)) + 1) + 3) + 1) + 3| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1cos(1\u03c0(1|(2x^2 + 3x + 3) + 3| + 2) - 2) + 3) - 2| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-2sin(2\u03c0((2(-1x + 4) + 2)/(2(-1x + 4) - 1)) - 3) - 3) + 1| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2|(3cos(3\u03c0(3x - 5) + 3) + 1) + 2| - 2)^2 + 4(-2|(3cos(3\u03c0(3x - 5) + 3) + 1) + 2| - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-4(1cos(3\u03c0((2x + 2)/(1x - 3))) - 2) - 5) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-2(2(-2cos(3\u03c0x - 3) + 1)^2 + 3(-2cos(3\u03c0x - 3) + 1) - 5)^2 + 2(2(-2cos(3\u03c0x - 3) + 1)^2 + 3(-2cos(3\u03c0x - 3) + 1) - 5) + 3)^2 + 5(-2(2(-2cos(3\u03c0x - 3) + 1)^2 + 3(-2cos(3\u03c0x - 3) + 1) - 5)^2 + 2(2(-2cos(3\u03c0x - 3) + 1)^2 + 3(-2cos(3\u03c0x - 3) + 1) - 5) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((-3(1(-3sin(2\u03c0x - 3) - 1)^2 + 3(-3sin(2\u03c0x - 3) - 1) + 5) - 2)/(2(1(-3sin(2\u03c0x - 3) - 1)^2 + 3(-3sin(2\u03c0x - 3) - 1) + 5) + 2))^2 - 2((-3(1(-3sin(2\u03c0x - 3) - 1)^2 + 3(-3sin(2\u03c0x - 3) - 1) + 5) - 2)/(2(1(-3sin(2\u03c0x - 3) - 1)^2 + 3(-3sin(2\u03c0x - 3) - 1) + 5) + 2)) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-2(1cos(2\u03c0(-1|x| + 1) + 2) - 1) - 2)/(3(1cos(2\u03c0(-1|x| + 1) + 2) - 1) - 2)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1|(-1sin(3\u03c0(-2|x - 1| - 2)) + 2) - 1| - 2)^2 + 4(1|(-1sin(3\u03c0(-2|x - 1| - 2)) + 2) - 1| - 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(2(1|(2|x - 1| - 3) + 1| - 2)^2 + 5(1|(2|x - 1| - 3) + 1| - 2) - 4) - 1)/(3(2(1|(2|x - 1| - 3) + 1| - 2)^2 + 5(1|(2|x - 1| - 3) + 1| - 2) - 4) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3|(-5(-3cos(3\u03c0x) - 3)^2 - 4(-3cos(3\u03c0x) - 3) - 3)|) + 1| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(2|((-2(1cos(1\u03c0x - 3) + 2) + 2)/(1(1cos(1\u03c0x - 3) + 2) + 2))| + 3) - 3| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-1((-3(5x^2 + 5x) + 3)/(-1(5x^2 + 5x) - 2))^2 + 2((-3(5x^2 + 5x) + 3)/(-1(5x^2 + 5x) - 2)) + 2)^2 - 1(-1((-3(5x^2 + 5x) + 3)/(-1(5x^2 + 5x) - 2))^2 + 2((-3(5x^2 + 5x) + 3)/(-1(5x^2 + 5x) - 2)) + 2) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1((-1(3(-2sin(3\u03c0x + 3) + 1)^2 + 2(-2sin(3\u03c0x + 3) + 1)) + 1)/(2(3(-2sin(3\u03c0x + 3) + 1)^2 + 2(-2sin(3\u03c0x + 3) + 1)) + 1)) + 1)/(2((-1(3(-2sin(3\u03c0x + 3) + 1)^2 + 2(-2sin(3\u03c0x + 3) + 1)) + 1)/(2(3(-2sin(3\u03c0x + 3) + 1)^2 + 2(-2sin(3\u03c0x + 3) + 1)) + 1)) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-1|(3(-2sin(2\u03c0x + 3) + 3)^2 - 1(-2sin(2\u03c0x + 3) + 3) - 2) - 2| - 1) - 3)/(2(-1|(3(-2sin(2\u03c0x + 3) + 3)^2 - 1(-2sin(2\u03c0x + 3) + 3) - 2) - 2| - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3cos(3\u03c0(-5((2x - 3)/(1x + 3)) + 1) - 2) - 3) - 2)/(1(3cos(3\u03c0(-5((2x - 3)/(1x + 3)) + 1) - 2) - 3)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2((1(5x^2 - 2x - 3))/(-2(5x^2 - 2x - 3) - 3)) + 4)^2 - 5(2((1(5x^2 - 2x - 3))/(-2(5x^2 - 2x - 3) - 3)) + 4) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3cos(2\u03c0(5(3x^2 - 5) + 2) + 2) - 3) + 2| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1sin(2\u03c0(-1|(-1|x + 3|) - 1| - 3) + 3) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(2(5(1cos(2\u03c0x - 1))^2 + 1(1cos(2\u03c0x - 1))) - 2)^2 + 4(2(5(1cos(2\u03c0x - 1))^2 + 1(1cos(2\u03c0x - 1))) - 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-2(-4(2sin(1\u03c0x + 3))^2 + 1(2sin(1\u03c0x + 3)))^2 + 2(-4(2sin(1\u03c0x + 3))^2 + 1(2sin(1\u03c0x + 3))) - 2) + 3)/(3(-2(-4(2sin(1\u03c0x + 3))^2 + 1(2sin(1\u03c0x + 3)))^2 + 2(-4(2sin(1\u03c0x + 3))^2 + 1(2sin(1\u03c0x + 3))) - 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(2(-2sin(2\u03c0(1x^2 + 4x - 5) + 3) + 3)^2 - 5(-2sin(2\u03c0(1x^2 + 4x - 5) + 3) + 3)) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2(4(2cos(1\u03c0x - 3) + 1)^2 - 1(2cos(1\u03c0x - 3) + 1) + 1)^2 - 5(4(2cos(1\u03c0x - 3) + 1)^2 - 1(2cos(1\u03c0x - 3) + 1) + 1) + 4)| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-1(-1|(-2|x| + 3) + 3| + 1)^2 + 5(-1|(-2|x| + 3) + 3| + 1) + 5) - 2| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-4(2cos(3\u03c0((-2x + 3)/(-2x - 3)) - 3) - 1) + 3) + 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2((-1(2sin(2\u03c0(-1x + 1) - 1) + 1) - 3)/(3(2sin(2\u03c0(-1x + 1) - 1) + 1) - 1)) + 3)/(1((-1(2sin(2\u03c0(-1x + 1) - 1) + 1) - 3)/(3(2sin(2\u03c0(-1x + 1) - 1) + 1) - 1)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((-2((-3(2cos(2\u03c0x - 2) - 1) - 3)/(-2(2cos(2\u03c0x - 2) - 1) - 3)) - 3)/(-1((-3(2cos(2\u03c0x - 2) - 1) - 3)/(-2(2cos(2\u03c0x - 2) - 1) - 3)) - 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-1sin(3\u03c0(1(1x - 1)^2 + 3(1x - 1) - 5) - 2) + 1) - 3)/(-1(-1sin(3\u03c0(1(1x - 1)^2 + 3(1x - 1) - 5) - 2) + 1) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((-2(1|(-4x^2 - 2x + 2) - 1| + 1) + 1)/(-3(1|(-4x^2 - 2x + 2) - 1| + 1) + 4)) - 2| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1(1sin(3\u03c0(-2|x + 2| - 3) - 2) + 3) + 1) + 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1|(-3cos(2\u03c0(4x - 3) - 2) + 1) - 2|) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((-3(-2(-2sin(3\u03c0x - 3) + 2) - 4) - 2)/(1(-2(-2sin(3\u03c0x - 3) + 2) - 4) + 1))| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-5(-2sin(2\u03c0(-3|x + 2| - 3) + 3) + 3) - 1) - 2| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(-5(-2|(-5x^2 + 4x) + 3| - 3) - 4) - 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1|(-2sin(1\u03c0(-4x^2 - 2x + 5) - 3) + 3) - 2| + 1)^2 + 4(1|(-2sin(1\u03c0(-4x^2 - 2x + 5) - 3) + 3) - 2| + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-1|(2cos(1\u03c0(-1|x + 3| - 1) + 3) + 2) + 3| - 2) + 2)/(-2(-1|(2cos(1\u03c0(-1|x + 3| - 1) + 3) + 2) + 3| - 2) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(5(-2sin(2\u03c0((1x - 3)/(2x + 3)) + 1) + 2) + 1) + 2| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(2|(1(3sin(3\u03c0x - 1) + 3)^2 + 1(3sin(3\u03c0x - 1) + 3) - 1) + 2| - 3)^2 + 3(2|(1(3sin(3\u03c0x - 1) + 3)^2 + 1(3sin(3\u03c0x - 1) + 3) - 1) + 2| - 3) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3cos(2\u03c0(-5((2x)/(2x - 1))^2 - 1((2x)/(2x - 1)) + 1) + 2) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2|(1cos(2\u03c0(2x - 3) - 2)) + 3| - 3)^2 - 5(2|(1cos(2\u03c0(2x - 3) - 2)) + 3| - 3) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2((3(-2cos(3\u03c0(1|x + 3|) - 2) - 2) + 3)/(1(-2cos(3\u03c0(1|x + 3|) - 2) - 2) + 2)) - 3)/(3((3(-2cos(3\u03c0(1|x + 3|) - 2) - 2) + 3)/(1(-2cos(3\u03c0(1|x + 3|) - 2) - 2) + 2)) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0((-1(-3|(-1x + 5) - 2| + 3) - 2)/(3(-3|(-1x + 5) - 2| + 3) + 3)) + 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1((-3((2(2cos(1\u03c0x - 1) - 3) - 2)/(-1(2cos(1\u03c0x - 1) - 3) - 2)))/(-1((2(2cos(1\u03c0x - 1) - 3) - 2)/(-1(2cos(1\u03c0x - 1) - 3) - 2)) - 2)) + 3)/(1((-3((2(2cos(1\u03c0x - 1) - 3) - 2)/(-1(2cos(1\u03c0x - 1) - 3) - 2)))/(-1((2(2cos(1\u03c0x - 1) - 3) - 2)/(-1(2cos(1\u03c0x - 1) - 3) - 2)) - 2))) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1|(-3cos(1\u03c0(2x^2 - 2x + 3) - 2) - 3) + 1| - 2) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(5(3cos(2\u03c0((1x + 1)/(-3x + 2)) - 3) + 1)^2 + 4(3cos(2\u03c0((1x + 1)/(-3x + 2)) - 3) + 1) - 5)^2 - 3(5(3cos(2\u03c0((1x + 1)/(-3x + 2)) - 3) + 1)^2 + 4(3cos(2\u03c0((1x + 1)/(-3x + 2)) - 3) + 1) - 5) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(2sin(1\u03c0(-3|(1x + 2)| - 3) - 2) + 1) - 3)/(2(2sin(1\u03c0(-3|(1x + 2)| - 3) - 2) + 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|((-2(1sin(2\u03c0(4x + 5) + 2) + 1) + 3)/(-3(1sin(2\u03c0(4x + 5) + 2) + 1) + 4)) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-3((3(-2cos(3\u03c0x - 1) - 2) + 3)/(-1(-2cos(3\u03c0x - 1) - 2) - 2))^2 - 5((3(-2cos(3\u03c0x - 1) - 2) + 3)/(-1(-2cos(3\u03c0x - 1) - 2) - 2)) + 2) - 2| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((2(2cos(3\u03c0((-2x - 3)/(3x + 2)) - 3) - 3) + 1)/(-1(2cos(3\u03c0((-2x - 3)/(3x + 2)) - 3) - 3))) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(1(-3|(-2|x - 3| - 2) - 1| - 2)^2 - 3(-3|(-2|x - 3| - 2) - 1| - 2) - 1) - 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(3|(3|(2sin(1\u03c0x) - 2) - 3| - 2)|) + 2)/(-1(3|(3|(2sin(1\u03c0x) - 2) - 3| - 2)|) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(2\u03c0((-2(-1|(2x - 1) - 1| + 1) - 2)/(3(-1|(2x - 1) - 1| + 1) + 1)) - 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-2|(2(2sin(1\u03c0x - 2) - 3) + 3) - 1| - 1) + 1)/(1(-2|(2(2sin(1\u03c0x - 2) - 3) + 3) - 1| - 1) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2sin(2\u03c0(-3|((1x)/(-1x - 1)) + 2| + 3) - 3) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2sin(1\u03c0(3(2|x - 3| + 3) + 2) - 3))^2 - 2(-2sin(1\u03c0(3(2|x - 3| + 3) + 2) - 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(1(-1(-4x - 5)^2 - 4(-4x - 5) - 2) + 2)) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0(4(-5(-1|x + 1| + 2)^2 - 3(-1|x + 1| + 2) + 3)^2 - 1(-5(-1|x + 1| + 2)^2 - 3(-1|x + 1| + 2) + 3) - 1) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1cos(3\u03c0(-2(-1|x - 1| - 3)^2 + 1) + 1) + 2)^2 + 5(1cos(3\u03c0(-2(-1|x - 1| - 3)^2 + 1) + 1) + 2) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(3|(-3sin(1\u03c0(-5x^2 + 3x + 5) + 2) - 2) - 1| + 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2sin(2\u03c0(-1(-2|x - 3|) - 1)) + 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-2|(-3|(-2x - 2) - 3| - 1) - 3| - 1) - 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2((3(1(-1sin(3\u03c0x - 1) - 1)^2 - 4(-1sin(3\u03c0x - 1) - 1) - 3) + 2)/(3(1(-1sin(3\u03c0x - 1) - 1)^2 - 4(-1sin(3\u03c0x - 1) - 1) - 3) - 2)) + 3)/(3((3(1(-1sin(3\u03c0x - 1) - 1)^2 - 4(-1sin(3\u03c0x - 1) - 1) - 3) + 2)/(3(1(-1sin(3\u03c0x - 1) - 1)^2 - 4(-1sin(3\u03c0x - 1) - 1) - 3) - 2)) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(3cos(3\u03c0(1(2|x - 2| + 3) - 3) + 2) + 3)| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((2(-1cos(3\u03c0((3x - 3)/(-2x + 3)) + 2) + 3) - 2)/(-2(-1cos(3\u03c0((3x - 3)/(-2x + 3)) + 2) + 3) + 3)) + 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-2|(1(-2cos(3\u03c0x - 1) + 2) + 5) + 2| - 2) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2cos(3\u03c0(5((2x)/(1x + 3))^2 - 1((2x)/(1x + 3)) - 3) + 3))^2 + 5(-2cos(3\u03c0(5((2x)/(1x + 3))^2 - 1((2x)/(1x + 3)) - 3) + 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-5(-1sin(1\u03c0((-1x - 2)/(1x)) + 2) + 2) + 3) + 3| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3sin(2\u03c0(4((-2x - 3)/(-1x - 2)) - 5) - 1) - 3)^2 + 1(-3sin(2\u03c0(4((-2x - 3)/(-1x - 2)) - 5) - 1) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1\u03c0(1|(-2(1x^2 + 2x + 5)^2 + 2(1x^2 + 2x + 5) + 2) - 1| + 1) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-4(1cos(3\u03c0((1x + 2)/(-1x)) + 2) - 3) + 5)^2 + 2(-4(1cos(3\u03c0((1x + 2)/(-1x)) + 2) - 3) + 5) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-1cos(2\u03c0(-3(-5x^2 - 5x - 5)^2 + 1(-5x^2 - 5x - 5) + 2) - 1)) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-2(-1cos(3\u03c0(-1x - 4) - 2) - 3)^2 + 3) - 3)/(-1(-2(-1cos(3\u03c0(-1x - 4) - 2) - 3)^2 + 3)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(1\u03c0(2|(2|(3x + 5) + 3| + 1) - 1| + 2) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(2|(-1cos(3\u03c0(-5x + 1) + 3) + 3) + 2| - 3)^2 - 3(2|(-1cos(3\u03c0(-5x + 1) + 3) + 3) + 2| - 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((2(-3sin(3\u03c0((1x - 1)/(2x + 2))) + 3) + 3)/(-3(-3sin(3\u03c0((1x - 1)/(2x + 2))) + 3) - 1)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1sin(2\u03c0(-2|(2x + 2) + 1| + 2) - 3) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3sin(2\u03c0(1|((-2x + 3)/(2x + 3)) - 2| + 3) - 2) + 1) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-1(1cos(3\u03c0(3x^2 + 2x + 2) + 3) - 2) + 2) + 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2|(-3sin(3\u03c0((2x - 1)/(1x)) + 2) + 2) - 1| + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1(-1|(1sin(2\u03c0x + 3) - 3) + 3|) + 2)^2 - 2(-1(-1|(1sin(2\u03c0x + 3) - 3) + 3|) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(5(-3|(1cos(2\u03c0x + 1) - 1) + 2| + 1) - 2) - 2)/(2(5(-3|(1cos(2\u03c0x + 1) - 1) + 2| + 1) - 2) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((-1(-3(1sin(1\u03c0x + 2) + 3) - 1) - 2)/(-1(-3(1sin(1\u03c0x + 2) + 3) - 1) + 3)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(3cos(2\u03c0((1((2x - 2)/(3x + 1)) + 3)/(2((2x - 2)/(3x + 1)) - 1)) + 1) - 1) - 1| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(5(2sin(1\u03c0(-2|x|) - 1)) + 3) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1|(-2(-2cos(3\u03c0x - 3) - 2) + 3)| - 2)^2 - 5(1|(-2(-2cos(3\u03c0x - 3) - 2) + 3)| - 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(1|(2(-1cos(3\u03c0x - 1) + 1) - 1) + 1| - 3)^2 + 4(1|(2(-1cos(3\u03c0x - 1) + 1) - 1) + 1| - 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(-3((3(-3x^2 - 2x + 1) - 2)/(-1(-3x^2 - 2x + 1) + 2)) - 1) + 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1(2sin(1\u03c0(1|x - 2| - 3) - 1) + 2) + 1) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1(5(-2cos(3\u03c0x - 1) + 2) + 3) - 1) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0(1(3(2x^2 - 4x - 4)^2 - 5(2x^2 - 4x - 4) - 4) + 2) - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(2(-1cos(2\u03c0(-2|x| - 3) + 2) + 3) - 1)^2 - 3(2(-1cos(2\u03c0(-2|x| - 3) + 2) + 3) - 1) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2(1(-1x^2 + 5x - 3)^2 + 3(-1x^2 + 5x - 3) - 4)^2 - 5(1(-1x^2 + 5x - 3)^2 + 3(-1x^2 + 5x - 3) - 4) + 2)^2 - 2(-2(1(-1x^2 + 5x - 3)^2 + 3(-1x^2 + 5x - 3) - 4)^2 - 5(1(-1x^2 + 5x - 3)^2 + 3(-1x^2 + 5x - 3) - 4) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(2cos(1\u03c0(-1(2|x + 2| + 3)^2 + 4) - 2) - 2) - 1)/(2(2cos(1\u03c0(-1(2|x + 2| + 3)^2 + 4) - 2) - 2) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2(-3(-3cos(1\u03c0x + 3))^2 + 2(-3cos(1\u03c0x + 3)) - 1)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(1|(-3|(-3cos(3\u03c0x + 2) - 1) + 1| - 1) + 3| - 3) + 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3cos(1\u03c0(4(-1|x - 2| - 3) - 4) + 3))^2 + 2(3cos(1\u03c0(4(-1|x - 2| - 3) - 4) + 3)) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-1sin(3\u03c0((1(1|x - 1| - 2) + 2)/(2(1|x - 1| - 2) - 1)) - 2) - 1)^2 + 5(-1sin(3\u03c0((1(1|x - 1| - 2) + 2)/(2(1|x - 1| - 2) - 1)) - 2) - 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-2cos(2\u03c0(1((-3x - 2)/(-1x - 3))^2 + 2((-3x - 2)/(-1x - 3)) - 3) + 2) + 3)| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1sin(1\u03c0(-4((-2x + 1)/(2x + 3))^2 - 2((-2x + 1)/(2x + 3)) + 3) + 2) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0((-1(5((3x + 3)/(-3x + 2))^2 - 4((3x + 3)/(-3x + 2)) - 1) - 3)/(-2(5((3x + 3)/(-3x + 2))^2 - 4((3x + 3)/(-3x + 2)) - 1) - 1)) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-2cos(3\u03c0(-2((-2x + 2)/(2x + 2))^2 - 5((-2x + 2)/(2x + 2)) - 5)) + 3) + 1)/(1(-2cos(3\u03c0(-2((-2x + 2)/(2x + 2))^2 - 5((-2x + 2)/(2x + 2)) - 5)) + 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-2|(-2cos(3\u03c0x - 1) - 3)| + 2)^2 + 2(-2|(-2cos(3\u03c0x - 1) - 3)| + 2) + 1)^2 + 3(1(-2|(-2cos(3\u03c0x - 1) - 3)| + 2)^2 + 2(-2|(-2cos(3\u03c0x - 1) - 3)| + 2) + 1) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2(3sin(1\u03c0(-1|x + 2| - 1) + 2) + 2)^2 + 2(3sin(1\u03c0(-1|x + 2| - 1) + 2) + 2) - 4) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(1cos(2\u03c0(3(-3|x + 1| + 2)^2 + 3(-3|x + 1| + 2) + 2) - 3)) + 3)/(2(1cos(2\u03c0(3(-3|x + 1| + 2)^2 + 3(-3|x + 1| + 2) + 2) - 3)) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(-3|(-5((-1x - 3)/(-1x))^2 + 4((-1x - 3)/(-1x))) + 1| + 2)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(2|(2cos(2\u03c0(-3|x| - 3) - 1) - 3) + 1| + 2)| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(2(-3(2x^2 - 4x - 3) - 5) - 3) - 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(1\u03c0(-3((1((2x + 3)/(-3x + 2)))/(2((2x + 3)/(-3x + 2)) + 1))^2 - 2)) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1|(3cos(1\u03c0(-1x) - 2)) - 2| - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((3(2sin(3\u03c0(-1|x + 3| + 2) + 3) + 1) - 3)/(-1(2sin(3\u03c0(-1|x + 3| + 2) + 3) + 1) - 1))^2 - 1((3(2sin(3\u03c0(-1|x + 3| + 2) + 3) + 1) - 3)/(-1(2sin(3\u03c0(-1|x + 3| + 2) + 3) + 1) - 1)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1(-1sin(1\u03c0((2x)/(1x + 3)) + 3) - 2)^2 + 1(-1sin(1\u03c0((2x)/(1x + 3)) + 3) - 2) + 5) - 1)/(-3(-1(-1sin(1\u03c0((2x)/(1x + 3)) + 3) - 2)^2 + 1(-1sin(1\u03c0((2x)/(1x + 3)) + 3) - 2) + 5) + 4) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(2|((-1(1|x - 3| - 1) + 2)/(2(1|x - 3| - 1) - 3)) - 2| - 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2((1(-3cos(1\u03c0x - 1) - 3) + 2)/(3(-3cos(1\u03c0x - 1) - 3) + 2)) - 2)^2 - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((1(2|(3cos(2\u03c0x - 1) + 1) - 2| - 2) + 2)/(3(2|(3cos(2\u03c0x - 1) + 1) - 2| - 2) + 2)) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0((-3(-3((1x + 1)/(-1x + 2))^2 + 3((1x + 1)/(-1x + 2)) + 2) + 3)/(-1(-3((1x + 1)/(-1x + 2))^2 + 3((1x + 1)/(-1x + 2)) + 2) + 3)) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-5(-1sin(1\u03c0(1|x + 1| - 1) + 3) - 1) - 3)^2 - 1(-5(-1sin(1\u03c0(1|x + 1| - 1) + 3) - 1) - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0((2(3|(-3|x - 1| + 2) + 3|) + 3)/(3(3|(-3|x - 1| + 2) + 3|) + 2)) - 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((-1(-4(-1cos(1\u03c0x - 1)) - 4) - 1)/(1(-4(-1cos(1\u03c0x - 1)) - 4) - 2))^2 - 4((-1(-4(-1cos(1\u03c0x - 1)) - 4) - 1)/(1(-4(-1cos(1\u03c0x - 1)) - 4) - 2)) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-2|(1(-2sin(3\u03c0x - 2) - 2) - 1) + 1| - 1) - 3| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(1\u03c0(-3|((3(5x - 1))/(1(5x - 1) - 3)) - 3| + 2) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2\u03c0(-3|(-3|(-1x^2 - 2x - 3) - 2| - 3) + 1| - 1) + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(2(3(-1sin(3\u03c0x + 2) + 3)^2 - 5(-1sin(3\u03c0x + 2) + 3) - 2)^2 - 5(3(-1sin(3\u03c0x + 2) + 3)^2 - 5(-1sin(3\u03c0x + 2) + 3) - 2) - 2) - 2)/(3(2(3(-1sin(3\u03c0x + 2) + 3)^2 - 5(-1sin(3\u03c0x + 2) + 3) - 2)^2 - 5(3(-1sin(3\u03c0x + 2) + 3)^2 - 5(-1sin(3\u03c0x + 2) + 3) - 2) - 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1sin(1\u03c0((-2(-5x + 4) + 2)/(-2(-5x + 4) - 3)) + 1) - 1)| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-2cos(2\u03c0(-3|(-2|x - 1| + 2)| + 1) - 2) - 1) + 2| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(2cos(2\u03c0(1(1x - 4)^2 + 1(1x - 4) + 4) - 1) + 2)| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1|(1(-4x^2 - 4x - 4) + 3) + 2| + 1)^2 - 2(-1|(1(-4x^2 - 4x - 4) + 3) + 2| + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0((2(-1(2|x - 1|) + 3) - 3)/(-1(-1(2|x - 1|) + 3) + 2)) + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-2|(1cos(2\u03c0(-1x - 1) + 3) + 2) + 3| + 1) - 3)/(-2(-2|(1cos(2\u03c0(-1x - 1) + 3) + 2) + 3| + 1) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-2cos(3\u03c0((2(3|x - 2| - 3) - 1)/(1(3|x - 2| - 3)))))| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(2(-3|(2x) - 1|)^2 + 4(-3|(2x) - 1|) - 3) - 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-1(-3sin(2\u03c0(-1x^2 - 3x - 3) + 3) - 3)^2 - 2(-3sin(2\u03c0(-1x^2 - 3x - 3) + 3) - 3) - 4) + 2)/(-3(-1(-3sin(2\u03c0(-1x^2 - 3x - 3) + 3) - 3)^2 - 2(-3sin(2\u03c0(-1x^2 - 3x - 3) + 3) - 3) - 4)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-1(1(1sin(3\u03c0x - 2) + 2) + 2) - 3) - 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1(2cos(3\u03c0(-1x^2 - 1x + 3)) + 1) + 1) + 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0(-2(2(1x - 3) + 4) - 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2cos(3\u03c0(2|(4x^2 - 5x - 1) + 3| - 3) + 3))^2 + 5(-2cos(3\u03c0(2|(4x^2 - 5x - 1) + 3| - 3) + 3)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(-3(-1|x - 1| + 2) + 5) - 3| + 1)^2 + 1(-1|(-3(-1|x - 1| + 2) + 5) - 3| + 1) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((3(-3sin(3\u03c0(3x - 1)) - 1) + 3)/(2(-3sin(3\u03c0(3x - 1)) - 1) - 3)) + 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(2|(1(1x^2 + 1x + 3) + 1) - 1| - 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2sin(2\u03c0((3(-3x - 4) - 1)/(-1(-3x - 4) - 2)) - 3) + 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1(2sin(3\u03c0(1x^2 - 3))) + 2)^2 + 5(-1(2sin(3\u03c0(1x^2 - 3))) + 2) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3|(-1sin(1\u03c0(-5x - 1) + 2) - 1) + 2| - 1))/(2(-3|(-1sin(1\u03c0(-5x - 1) + 2) - 1) + 2| - 1) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1|(5(1|x - 1| - 2)^2 - 3(1|x - 1| - 2) - 5) - 1| + 2) + 2)/(-1(-1|(5(1|x - 1| - 2)^2 - 3(1|x - 1| - 2) - 5) - 1| + 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-2(-3sin(1\u03c0(1x) - 2) + 1) + 3)/(-2(-3sin(1\u03c0(1x) - 2) + 1)))^2 - 1((-2(-3sin(1\u03c0(1x) - 2) + 1) + 3)/(-2(-3sin(1\u03c0(1x) - 2) + 1))) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(2cos(1\u03c0(-5((-2x - 3)/(1x + 3))^2 + 5((-2x - 3)/(1x + 3)) + 5) + 1)) + 3)/(-3(2cos(1\u03c0(-5((-2x - 3)/(1x + 3))^2 + 5((-2x - 3)/(1x + 3)) + 5) + 1)) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}