{"messages": [{"content": "For the function f(x) = -2|(3(-3sin(2\u03c0x + 3) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(1cos(2\u03c0(1|(-5x^2 - 1x) + 1| - 2)) + 1) - 2)/(1(1cos(2\u03c0(1|(-5x^2 - 1x) + 1| - 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": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(2sin(3\u03c0(-1x - 4) - 3) + 3) - 2| - 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) = -2sin(1\u03c0(4((-2x)/(-1x - 3))^2 + 3((-2x)/(-1x - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(-2|(1x + 1) + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2cos(1\u03c0((2x)/(-1x - 2)) - 1)) + 2| - 1 is strictly increasing.\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) = 3(1sin(3\u03c0(-1|x + 3| - 2)) - 3)^2 + 4(1sin(3\u03c0(-1|x + 3| - 2)) - 3) - 2 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) = -2|((-3(1sin(3\u03c0(2x - 2) - 2) + 1) + 3)/(-2(1sin(3\u03c0(2x - 2) - 2) + 1) - 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": "62", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((2(-1sin(1\u03c0x + 3) + 2))/(-2(-1sin(1\u03c0x + 3) + 2) + 3))^2 + 4((2(-1sin(1\u03c0x + 3) + 2))/(-2(-1sin(1\u03c0x + 3) + 2) + 3)) - 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) = 3cos(3\u03c0(-3((-3(2|x + 2| + 2) - 1)/(-1(2|x + 2| + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3cos(3\u03c0(-5(-4x^2 + 3x - 3) - 2) - 3) - 2) - 1| + 2 is strictly decreasing.\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(-2sin(1\u03c0(3x - 5) - 1) + 1) + 3)/(-2(-2sin(1\u03c0(3x - 5) - 1) + 1) + 1) 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) = 5(1sin(3\u03c0(-3|((-3x + 2)/(2x + 3)) + 2| + 3) + 3))^2 - 3(1sin(3\u03c0(-3|((-3x + 2)/(2x + 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": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((-2(-1cos(1\u03c0(2|x - 2| - 1) - 3) - 1) - 3)/(1(-1cos(1\u03c0(2|x - 2| - 1) - 3) - 1) + 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) = 3((-1(-2sin(1\u03c0(2x - 5) + 1) - 1) - 2)/(3(-2sin(1\u03c0(2x - 5) + 1) - 1) + 1))^2 + 2((-1(-2sin(1\u03c0(2x - 5) + 1) - 1) - 2)/(3(-2sin(1\u03c0(2x - 5) + 1) - 1) + 1)) + 3 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) = -2cos(3\u03c0(4(2|x| - 2) - 1) + 2) + 3 positive?\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) = 2(-4(1sin(2\u03c0x - 3) + 2) + 2)^2 - 4(-4(1sin(2\u03c0x - 3) + 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": "For the function f(x) = -1(1cos(2\u03c0(1((-2x + 1)/(2x + 1)) + 5) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2\u03c0(5(1|x - 1| + 1)^2 - 3(1|x - 1| + 1) - 1) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-2cos(1\u03c0(5x + 2) - 2) + 3) - 2)/(3(-2cos(1\u03c0(5x + 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": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(-4((-1x + 1)/(3x - 2)) + 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) = 1sin(1\u03c0(-2|(-1x + 5) - 1| - 2)) + 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) = 1cos(3\u03c0(1|(5(4x^2 - 2x + 5) + 5) + 1| - 2) + 2) + 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) = 3|(-2cos(1\u03c0(-4x^2 - 5x) - 1) + 3) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-3cos(2\u03c0(1|x - 1| - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2(-2cos(1\u03c0x - 3) + 3) + 2) - 2)/(-3(-2(-2cos(1\u03c0x - 3) + 3) + 2)) 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) = -2(3|(-1sin(2\u03c0x + 3) - 1) - 2|) - 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) = -3sin(1\u03c0(1(1((3x - 3)/(-1x + 3)) + 1)^2 - 5(1((3x - 3)/(-1x + 3)) + 1) + 5) - 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(1|(-5(-2cos(1\u03c0x + 3))^2 + 4(-2cos(1\u03c0x + 3)) + 3) + 3|))/(3(1|(-5(-2cos(1\u03c0x + 3))^2 + 4(-2cos(1\u03c0x + 3)) + 3) + 3|) + 3) 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) = 2|((-2(3cos(1\u03c0x + 3) + 3) - 1)/(-1(3cos(1\u03c0x + 3) + 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) = -2|(-4(1cos(3\u03c0(-3x - 3) - 2))^2 - 5(1cos(3\u03c0(-3x - 3) - 2)) - 1) - 2| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-3sin(1\u03c0(2x - 5) + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-2cos(3\u03c0((1(2|x + 2|) - 3)/(2(2|x + 2|) - 1)) + 3) + 1)^2 + 5(-2cos(3\u03c0((1(2|x + 2|) - 3)/(2(2|x + 2|) - 1)) + 3) + 1) + 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) = 1sin(3\u03c0((-3(-4x^2 - 4x - 3) - 1)/(1(-4x^2 - 4x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-3(-2sin(2\u03c0(4x)) + 3)^2 + 5(-2sin(2\u03c0(4x)) + 3) - 1) + 2)/(-3(-3(-2sin(2\u03c0(4x)) + 3)^2 + 5(-2sin(2\u03c0(4x)) + 3) - 1) + 2) 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) = (2(-3cos(3\u03c0(1|x + 1| + 2) + 2) + 3))/(-2(-3cos(3\u03c0(1|x + 1| + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((2(4(-2cos(3\u03c0x + 3) + 3) + 4) + 3)/(1(4(-2cos(3\u03c0x + 3) + 3) + 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": "For the function f(x) = 2sin(3\u03c0(1((3x + 3)/(-3x + 1))^2 + 5((3x + 3)/(-3x + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-1|(2sin(3\u03c0x + 1) - 3) + 3| - 2) + 2)/(-3(-1|(2sin(3\u03c0x + 1) - 3) + 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) = 5(-1sin(2\u03c0(-1|x + 3| + 1) + 2) + 1) - 3 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(-5(-1cos(2\u03c0x - 2) - 3) + 1)^2 - 3(-5(-1cos(2\u03c0x - 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3((3(3sin(1\u03c0x - 2) + 3) - 3)/(-3(3sin(1\u03c0x - 2) + 3))) - 4)^2 + 5(3((3(3sin(1\u03c0x - 2) + 3) - 3)/(-3(3sin(1\u03c0x - 2) + 3))) - 4) 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) = -1sin(2\u03c0(3(-2|x + 1|)^2 - 1(-2|x + 1|) - 3) - 1) + 1 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) = -4((-1(-3sin(3\u03c0(-3|x + 2| + 2))) + 2)/(3(-3sin(3\u03c0(-3|x + 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": "77", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((1(-2cos(1\u03c0x + 3) + 1) + 1)/(-2(-2cos(1\u03c0x + 3) + 1) + 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) = 1|(5(3sin(3\u03c0x - 2) - 2)^2 - 5(3sin(3\u03c0x - 2) - 2) - 3) + 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) = 3cos(2\u03c0(-4((3(-2|x| + 3) - 2)/(1(-2|x| + 3) + 2)) - 5) - 2) - 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((3(3cos(3\u03c0(4x^2 + 5x - 3) + 2) - 3) - 1)/(2(3cos(3\u03c0(4x^2 + 5x - 3) + 2) - 3))) - 1 negative?\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) = -1cos(2\u03c0(-5(1(2|x| - 3)^2 + 1(2|x| - 3) + 3) + 1) - 1) - 1 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(-2|((-2(-2cos(3\u03c0x - 2) + 2) - 2)/(-3(-2cos(3\u03c0x - 2) + 2) - 2)) - 1| - 3) + 4 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) = -4(-1(1sin(1\u03c0(1|x - 3|) - 3) + 2) - 3)^2 + 4(-1(1sin(1\u03c0(1|x - 3|) - 3) + 2) - 3) + 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) = -5(-3cos(3\u03c0(1((3x - 3)/(-1x + 2))^2 - 3((3x - 3)/(-1x + 2)) - 2) + 3) + 1) + 4 positive?\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) = 2(-3|(-3(3cos(1\u03c0x - 1) + 2)^2 - 5)| - 2) - 1 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) = -2cos(3\u03c0((-2(2|(2x^2 - 5x) - 1|) + 1)/(1(2|(2x^2 - 5x) - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2|((-3(2cos(3\u03c0x - 3) - 3) + 2)/(-2(2cos(3\u03c0x - 3) - 3) - 2)) - 3| + 3)^2 + 3(2|((-3(2cos(3\u03c0x - 3) - 3) + 2)/(-2(2cos(3\u03c0x - 3) - 3) - 2)) - 3| + 3) + 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) = (1(-2|(2cos(1\u03c0x + 1) - 3)| + 3) + 2)/(-1(-2|(2cos(1\u03c0x + 1) - 3)| + 3) - 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) = -3|((-1(1(2cos(1\u03c0x - 1) + 1)^2 + 3) + 1)/(-3(1(2cos(1\u03c0x - 1) + 1)^2 + 3) + 2)) + 1| + 2 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((-2(-2sin(1\u03c0x) + 3) + 2)/(-2(-2sin(1\u03c0x) + 3) - 1)) - 1 negative?\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) = -1sin(3\u03c0(-2(-2|x - 2| - 2) + 3)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "85", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(5(-1sin(2\u03c0(-3|x + 1| + 1) - 2) - 3)^2 - 3(-1sin(2\u03c0(-3|x + 1| + 1) - 2) - 3) - 2))/(2(5(-1sin(2\u03c0(-3|x + 1| + 1) - 2) - 3)^2 - 3(-1sin(2\u03c0(-3|x + 1| + 1) - 2) - 3) - 2) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0(4((2(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2) - 3))^2 + 1((2(3|x + 2| + 2) + 2)/(1(3|x + 2| + 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) = 1cos(1\u03c0(2(-1(-1|x + 3| + 3))^2 - 4(-1(-1|x + 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": "67", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(-2(2|x + 1| + 1) + 5) - 1) positive?\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) = -1|((-2(-1cos(1\u03c0(-3x^2 + 4x + 2) + 1)) + 1)/(-1(-1cos(1\u03c0(-3x^2 + 4x + 2) + 1)) - 3)) - 1| + 3 is strictly decreasing.\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) = -1((-1(-1sin(3\u03c0(4x + 1) - 3)))/(1(-1sin(3\u03c0(4x + 1) - 3)) + 3))^2 - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1(2cos(3\u03c0(-2x^2 - 4x - 1) + 1) - 1) + 4)| - 1 negative?\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) = -2|(2cos(2\u03c0(4x^2 + 3) + 3) + 2) + 3| + 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) = 1(4(-2|(-2cos(3\u03c0x - 2) + 3) + 3| + 3)^2 + 5(-2|(-2cos(3\u03c0x - 2) + 3) + 3| + 3) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3(-1cos(3\u03c0(2|x - 1| - 3) - 3) - 2) + 2)/(3(-1cos(3\u03c0(2|x - 1| - 3) - 3) - 2) + 3))^2 - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "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 + 1) + 2)| + 3))/(2(3|(-2sin(2\u03c0x + 1) + 2)| + 3) + 1)) - 4 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) = 2((1(-3sin(1\u03c0x + 1) - 3) - 1)/(-1(-3sin(1\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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(1|((-3(-1x^2 + 3) + 2)/(-1(-1x^2 + 3) + 2)) + 2| + 1) + 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) = -1|(1cos(2\u03c0(2x - 2) + 1) - 3) + 2| - 3 negative?\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) = 3cos(3\u03c0((-1(-1(-2|x - 1| + 1)) + 3)/(3(-1(-2|x - 1| + 1)) + 2))) + 3 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) = 2|(3sin(2\u03c0(2x + 4) - 2) - 1) - 2| + 3 positive?\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(-1cos(3\u03c0(2|(-4x^2 + 1x - 4) - 2| - 1) - 3) - 2) - 1)/(-2(-1cos(3\u03c0(2|(-4x^2 + 1x - 4) - 2| - 1) - 3) - 2) - 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) = -1|(1sin(2\u03c0(1(1x^2 + 3) - 5) - 2) + 3) - 3| 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|(2sin(2\u03c0((1x + 2)/(-2x - 1)) + 3) + 3) + 2| + 1 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) = 3cos(3\u03c0((-3(1(-2x + 2)^2 - 5(-2x + 2) + 2) - 2)/(1(1(-2x + 2)^2 - 5(-2x + 2) + 2) + 3)) - 2) - 3 positive?\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|(-3sin(2\u03c0(-5x^2 + 1x)) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-3(-2sin(3\u03c0((2x + 1)/(-2x + 3)) + 3) + 3)^2 + 4(-2sin(3\u03c0((2x + 1)/(-2x + 3)) + 3) + 3)) - 2| - 1 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) = 2|(-5(5(1cos(1\u03c0x - 2) + 1) - 4)^2 + 2(5(1cos(1\u03c0x - 2) + 1) - 4) + 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": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(2sin(2\u03c0(1((-1x + 1)/(-1x))^2 - 2((-1x + 1)/(-1x)) - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1sin(3\u03c0(-4x^2 + 1) + 3) - 1) - 2| - 1 is strictly decreasing.\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(-5(1|(1cos(2\u03c0x) + 2)| + 2) - 3) - 1)/(2(-5(1|(1cos(2\u03c0x) + 2)| + 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) = 1(1(-2sin(2\u03c0x - 3) + 2)^2 + 5(-2sin(2\u03c0x - 3) + 2) - 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) = -2|(2cos(2\u03c0(5((3x - 1)/(-2x + 1))^2 - 4((3x - 1)/(-2x + 1))) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2sin(1\u03c0(-2((-3x - 3)/(3x + 2))^2 + 3((-3x - 3)/(3x + 2)) + 4) - 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((-2(5(-1|x - 1| + 2)^2 - 4(-1|x - 1| + 2) + 2) - 1)/(1(5(-1|x - 1| + 2)^2 - 4(-1|x - 1| + 2) + 2) + 2)) - 3) - 1 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) = -2sin(2\u03c0(-3(-1((-3x + 1)/(-1x))^2 + 5((-3x + 1)/(-1x)) + 4) + 1) - 3) - 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) = -2|(-4((-2(2cos(1\u03c0x - 1) - 1) - 2)/(-1(2cos(1\u03c0x - 1) - 1) + 2))^2 + 5((-2(2cos(1\u03c0x - 1) - 1) - 2)/(-1(2cos(1\u03c0x - 1) - 1) + 2)) - 4) + 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) = 2sin(2\u03c0((2(-3(2|x + 2| - 3)^2 - 3(2|x + 2| - 3) - 5) + 2)/(-1(-3(2|x + 2| - 3)^2 - 3(2|x + 2| - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(4(-2sin(3\u03c0(-3x + 3) + 3) - 1)^2 + 4(-2sin(3\u03c0(-3x + 3) + 3) - 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": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2(1sin(1\u03c0x + 1) + 3) - 4) - 2| - 2 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) = 4(-1cos(3\u03c0(-5x^2 - 1) - 1) - 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2cos(3\u03c0((1x + 3)/(3x - 2)) - 1) - 1)^2 + 5(-2cos(3\u03c0((1x + 3)/(3x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-5(2sin(1\u03c0x - 2) - 1) - 4)^2 - 1(-5(2sin(1\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(4(-4x - 5)^2 - 5(-4x - 5) - 5) + 1) - 1 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) = (-1(3(3cos(3\u03c0x + 3) + 3) - 4) - 1)/(-3(3(3cos(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-3|((3(-1sin(3\u03c0x + 1) - 2))/(-2(-1sin(3\u03c0x + 1) - 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": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2\u03c0(4(-1x - 1)^2 + 2(-1x - 1) - 5) - 2) + 1 negative?\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) = (-3(2cos(2\u03c0(-3x - 2) - 1) - 2))/(-2(2cos(2\u03c0(-3x - 2) - 1) - 2) + 1) 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) = -2|(-2sin(3\u03c0((-2(2x - 3) - 1)/(2(2x - 3) - 1)) - 2) - 2) - 2| - 3 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|((1(-2cos(1\u03c0x - 1) - 3) + 3)/(2(-2cos(1\u03c0x - 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": "For the function f(x) = (-1(1sin(3\u03c0(-4(-5x + 5)^2 + 3) - 1) + 1) + 3)/(3(1sin(3\u03c0(-4(-5x + 5)^2 + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(2|(-2cos(1\u03c0(1x^2 - 1x)) - 1) - 2| + 3) - 3)/(-2(2|(-2cos(1\u03c0(1x^2 - 1x)) - 1) - 2| + 3) + 3) positive?\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(-4x^2 + 5) + 1) + 2)) + 1| - 2 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) = -3((1(-2sin(3\u03c0x + 3) + 3))/(-2(-2sin(3\u03c0x + 3) + 3) + 3)) + 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) = -1cos(1\u03c0(4(-1|x + 2| - 2) - 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": "For the function f(x) = -5((3(-4(-1cos(2\u03c0x + 1) + 3)^2 + 4(-1cos(2\u03c0x + 1) + 3) - 4) + 3)/(-2(-4(-1cos(2\u03c0x + 1) + 3)^2 + 4(-1cos(2\u03c0x + 1) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2cos(2\u03c0((-2x - 1)/(-2x - 3)) - 2) + 2)^2 - 4(-2cos(2\u03c0((-2x - 1)/(-2x - 3)) - 2) + 2) + 1 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) = -3(-1|(-2sin(3\u03c0(-2x + 2) - 2)) - 3| - 3)^2 - 2(-1|(-2sin(3\u03c0(-2x + 2) - 2)) - 3| - 3) - 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) = 3sin(1\u03c0(-5((2x + 2)/(2x)) - 5) + 1) + 2 negative?\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|(4(2cos(3\u03c0x - 2) - 2)^2 - 3(2cos(3\u03c0x - 2) - 2) - 1) + 1|))/(3(-2|(4(2cos(3\u03c0x - 2) - 2)^2 - 3(2cos(3\u03c0x - 2) - 2) - 1) + 1|) + 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(1cos(3\u03c0(2|(3x - 3) - 3| + 2) - 1) + 1)^2 + 2 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((-1(-3|(-2sin(1\u03c0x + 2) + 3) + 1| - 3) - 2)/(2(-3|(-2sin(1\u03c0x + 2) + 3) + 1| - 3) + 1))^2 - 1((-1(-3|(-2sin(1\u03c0x + 2) + 3) + 1| - 3) - 2)/(2(-3|(-2sin(1\u03c0x + 2) + 3) + 1| - 3) + 1)) + 5 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(2cos(3\u03c0(-4(-1x - 3)^2 + 4(-1x - 3) - 4) + 1) + 1))/(3(2cos(3\u03c0(-4(-1x - 3)^2 + 4(-1x - 3) - 4) + 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(3cos(1\u03c0(3x + 5) - 3) + 3) - 1)/(2(3cos(1\u03c0(3x + 5) - 3) + 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) = -2cos(2\u03c0(1(1|x - 1| + 1)^2 - 5(1|x - 1| + 1) - 4) - 2) is strictly decreasing.\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) = 2|(5(-1cos(1\u03c0x + 1) + 2)) + 3| - 1 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) = -3sin(3\u03c0(-4(-3|((2x - 1)/(-1x + 2)) - 2|)^2 - 4(-3|((2x - 1)/(-1x + 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": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(1(1x + 4)^2 + 3(1x + 4) + 4) - 3) + 3 is strictly decreasing.\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) = -3|((1(-1cos(3\u03c0(1x - 3) + 1) - 1) + 2)/(-1(-1cos(3\u03c0(1x - 3) + 1) - 1) - 3))| - 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) = -3sin(2\u03c0(-1|(5x^2 + 1x + 2) - 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": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1((-2(-1cos(2\u03c0x - 1) + 3) + 3)/(-3(-1cos(2\u03c0x - 1) + 3) - 2))^2 + 1((-2(-1cos(2\u03c0x - 1) + 3) + 3)/(-3(-1cos(2\u03c0x - 1) + 3) - 2)) + 5) + 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) = 2sin(1\u03c0(-1|(-5x^2 - 3x - 5) - 2| - 3) + 1) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(3|(4(-4x^2 - 5x + 2) + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-3sin(1\u03c0(-3|x + 3| - 1)) + 2) - 3)/(-2(-3sin(1\u03c0(-3|x + 3| - 1)) + 2) + 1) 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) = -4((3(-3|(-2cos(2\u03c0x - 1) - 3) + 1| + 1) - 1)/(3(-3|(-2cos(2\u03c0x - 1) - 3) + 1| + 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": "For the function f(x) = -1cos(3\u03c0((-3(2|(2x - 5)| + 1) - 1)/(1(2|(2x - 5)| + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3(2sin(3\u03c0x - 3) - 1) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|((2(-1cos(3\u03c0x + 1) + 1) - 2)/(3(-1cos(3\u03c0x + 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": "30", "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 - 3)/(-2x - 3))^2 - 4((-1x - 3)/(-2x - 3))) - 1)) + 2| + 3 is strictly decreasing.\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(-3sin(3\u03c0(2|x| - 2)) - 3) - 3)/(-2(-3sin(3\u03c0(2|x| - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(-1|((-2x)/(-1x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-2sin(3\u03c0((-2(4x) - 2)/(-1(4x) - 3)) + 3) - 2)^2 - 3(-2sin(3\u03c0((-2(4x) - 2)/(-1(4x) - 3)) + 3) - 2) 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) = 1|(-1cos(2\u03c0((-3(2x - 4) + 3)/(1(2x - 4))))) - 3| 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) = 2|(5(-3cos(3\u03c0(-3x - 5)) + 3)^2 - 4(-3cos(3\u03c0(-3x - 5)) + 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": "17", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(3(3|x - 3|)^2 - 3(3|x - 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": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1cos(3\u03c0(1x^2 - 5) + 2) - 3) + 3)/(1(-1cos(3\u03c0(1x^2 - 5) + 2) - 3) - 2) positive?\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(2|(5(3sin(3\u03c0x + 1))^2 + 1(3sin(3\u03c0x + 1)) - 1) + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3sin(1\u03c0(3|x|) - 3)) + 1 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) = -2sin(2\u03c0(4(3x - 2)^2 + 1(3x - 2) - 2)) + 1 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|(2((-1(2cos(2\u03c0x - 1) - 3) + 3)/(-1(2cos(2\u03c0x - 1) - 3) + 2))^2 - 4((-1(2cos(2\u03c0x - 1) - 3) + 3)/(-1(2cos(2\u03c0x - 1) - 3) + 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) = -2sin(1\u03c0(-2|(-3x^2 + 1x + 2)| + 1) - 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(2(3(-3sin(1\u03c0x - 3))^2 - 5(-3sin(1\u03c0x - 3)) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-3(1sin(1\u03c0((2x + 2)/(-1x + 2)) - 1) + 1)^2 + 2(1sin(1\u03c0((2x + 2)/(-1x + 2)) - 1) + 1) - 5) - 2 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) = (-3(1|(-2cos(3\u03c0x + 1) + 3) + 2| - 2) + 1)/(-1(1|(-2cos(3\u03c0x + 1) + 3) + 2| - 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) = 2cos(3\u03c0(2|((-2x)/(3x + 2)) - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-5(-3(3cos(1\u03c0x - 2) - 1)^2 + 4) - 2) + 1)/(-2(-5(-3(3cos(1\u03c0x - 2) - 1)^2 + 4) - 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) = 2|(3sin(2\u03c0((1x + 3)/(2x - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-1sin(2\u03c0(-2|((2x)/(1x + 1)) - 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": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1(1sin(1\u03c0((-2x - 1)/(-1x + 3)) + 2) - 2)^2 - 1(1sin(1\u03c0((-2x - 1)/(-1x + 3)) + 2) - 2) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-3|(-2cos(2\u03c0(4x + 1)) - 1) + 1| - 3))/(-1(-3|(-2cos(2\u03c0(4x + 1)) - 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": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(-1((1(2x - 3) + 2)/(1(2x - 3)))^2 + 4) + 1) + 2 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) = 1((2(3sin(2\u03c0(1|x + 3| - 3) - 3) + 2) - 1)/(2(3sin(2\u03c0(1|x + 3| - 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) = 2|(-1(-2sin(3\u03c0x + 3) + 1)^2 + 1(-2sin(3\u03c0x + 3) + 1) + 5) + 3| - 1 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) = -2|(-1(1sin(3\u03c0x - 1) + 2) + 3) - 1| - 1 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) = (3(3|(3sin(1\u03c0(4x + 1)) - 3) - 1|) + 3)/(-3(3|(3sin(1\u03c0(4x + 1)) - 3) - 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) = -2|(1sin(1\u03c0(3x - 1) + 3)) - 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) = -5(3|(3sin(3\u03c0x + 1) - 3) + 2| - 3) - 1 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) = -2|(-2sin(1\u03c0(4x + 1) - 2) - 3) + 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) = 3cos(1\u03c0((2(3(-3|x + 3| + 1)^2 + 4(-3|x + 3| + 1) - 1) + 1)/(3(3(-3|x + 3| + 1)^2 + 4(-3|x + 3| + 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) = -5((-1(2sin(3\u03c0x + 3) - 3))/(1(2sin(3\u03c0x + 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) = 1cos(2\u03c0(4(-4((-3x + 3)/(3x - 1))^2 + 3((-3x + 3)/(3x - 1)) - 1) + 2) - 1) - 1 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) = 3(-2|(-1cos(1\u03c0x + 3) - 2) + 2| - 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) = -1|(-5(-3sin(1\u03c0x - 2) - 1)^2 + 5(-3sin(1\u03c0x - 2) - 1) - 4) - 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) = 1|(-3cos(1\u03c0(2x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-2cos(2\u03c0((-1x + 2)/(-2x)) - 3) - 3) - 1 is strictly increasing.\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) = 2sin(1\u03c0(-2|(4(5x^2 + 1x - 2) - 3) + 3|) - 3) 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|(2(3cos(1\u03c0(-4x - 3) - 3) - 3)^2 + 3) - 3| + 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) = 5((3(-1sin(3\u03c0x - 1) - 2) + 3)/(-1(-1sin(3\u03c0x - 1) - 2) - 3))^2 + 5((3(-1sin(3\u03c0x - 1) - 2) + 3)/(-1(-1sin(3\u03c0x - 1) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1cos(2\u03c0(-3((3x)/(-2x - 3)) - 4) + 1) + 2)^2 + 5 is strictly increasing.\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|(-1sin(1\u03c0(4((3x + 1)/(2x - 1))^2 + 4((3x + 1)/(2x - 1)) + 3)) + 3) + 3| - 2 is strictly increasing.\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(-1sin(3\u03c0(-3|(-1x^2 + 1x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(3(2|(-1cos(3\u03c0x + 3) + 2) + 1| + 2) - 1) - 3)/(2(3(2|(-1cos(3\u03c0x + 3) + 2) + 1| + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0(2|(1(-5x + 5)^2 - 1(-5x + 5) + 2) - 2|) - 2) - 1 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) = (1(1cos(3\u03c0(1(2|x + 2| - 2) - 2) + 2) + 3) - 2)/(3(1cos(3\u03c0(1(2|x + 2| - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-3|(2sin(1\u03c0(-4x^2 - 3x - 2) - 3) - 2)| + 2) + 2)/(-2(-3|(2sin(1\u03c0(-4x^2 - 3x - 2) - 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": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0(3|(-4(4x^2 - 5x - 5) - 4) + 3| + 1) - 3) negative?\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|(1sin(3\u03c0(-2x^2 + 2x + 4) - 1) + 2) + 3| 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) = (3(-2(2sin(3\u03c0(3|x - 2| - 2) - 2) - 1)^2 + 5(2sin(3\u03c0(3|x - 2| - 2) - 2) - 1) - 2) + 1)/(-1(-2(2sin(3\u03c0(3|x - 2| - 2) - 2) - 1)^2 + 5(2sin(3\u03c0(3|x - 2| - 2) - 2) - 1) - 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(1cos(1\u03c0(4(-3|x - 2|) - 2) + 1)) - 2)/(-1(1cos(1\u03c0(4(-3|x - 2|) - 2) + 1)) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3sin(3\u03c0(5(2|x + 2|) - 2) + 3) - 3) - 2)/(1(-3sin(3\u03c0(5(2|x + 2|) - 2) + 3) - 3) + 1) 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) = -5((-3(1cos(3\u03c0x - 2) + 2) - 2)/(-2(1cos(3\u03c0x - 2) + 2))) - 4 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) = -1((3(-2cos(1\u03c0(-3|x - 2| + 3) - 3) + 1) - 3)/(-1(-2cos(1\u03c0(-3|x - 2| + 3) - 3) + 1) - 2))^2 + 1((3(-2cos(1\u03c0(-3|x - 2| + 3) - 3) + 1) - 3)/(-1(-2cos(1\u03c0(-3|x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(3(-2|x - 2| + 1) - 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) = 2(-2sin(3\u03c0(1|(-5x - 1) + 2| + 2)) + 1)^2 + 4(-2sin(3\u03c0(1|(-5x - 1) + 2| + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-3(2(3cos(2\u03c0x + 2) - 1)^2 - 1(3cos(2\u03c0x + 2) - 1)) - 5)| - 1 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) = 1((-2(-3cos(1\u03c0(1x^2 - 3x - 4) - 3) - 2) - 1)/(-1(-3cos(1\u03c0(1x^2 - 3x - 4) - 3) - 2) + 3)) - 1 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) = 3|(4(1cos(1\u03c0x - 2) + 1)) + 1| - 2 negative?\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(2\u03c0(2((3x + 3)/(-2x - 3))^2 - 2((3x + 3)/(-2x - 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": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-1sin(1\u03c0(-1|x - 3| - 1) - 1)) - 2)/(1(-1sin(1\u03c0(-1|x - 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) = -3cos(1\u03c0(-4(-1|((1x + 2)/(2x))| + 3)^2 + 5(-1|((1x + 2)/(2x))| + 3) + 4) - 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) = 2((1(-2|(3cos(3\u03c0x + 1) - 3) - 1| + 3) - 2)/(-2(-2|(3cos(3\u03c0x + 1) - 3) - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1sin(2\u03c0((2x - 1)/(-1x + 2)) - 2) - 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) = 3((2(1cos(1\u03c0x + 3)))/(-2(1cos(1\u03c0x + 3)) - 1))^2 + 4((2(1cos(1\u03c0x + 3)))/(-2(1cos(1\u03c0x + 3)) - 1)) - 4 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) = 1sin(1\u03c0(-3(-3(-2|x + 3| + 2) + 4)^2 + 5(-3(-2|x + 3| + 2) + 4) + 2) - 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(5(-2sin(3\u03c0x + 2) - 1) - 1)^2 + 3(5(-2sin(3\u03c0x + 2) - 1) - 1) + 4 positive?\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(5(2cos(1\u03c0x - 2) - 2)^2 - 3(2cos(1\u03c0x - 2) - 2) - 5) + 3)/(-2(5(2cos(1\u03c0x - 2) - 2)^2 - 3(2cos(1\u03c0x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1|(2cos(2\u03c0x - 3) - 3) + 3| + 1)^2 - 1(1|(2cos(2\u03c0x - 3) - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(2sin(2\u03c0(-1|(4x^2 - 2x - 5)| - 3) - 1) + 3) + 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) = -2((1(2(-1cos(2\u03c0x - 3) + 3)^2 - 1(-1cos(2\u03c0x - 3) + 3) - 5))/(-3(2(-1cos(2\u03c0x - 3) + 3)^2 - 1(-1cos(2\u03c0x - 3) + 3) - 5) - 2)) + 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) = -3|(3(-3cos(2\u03c0(-3x + 1) - 2) - 3)^2 - 5(-3cos(2\u03c0(-3x + 1) - 2) - 3) + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(2(-3sin(2\u03c0((-1x + 1)/(3x + 2)) - 3) - 3)^2 + 1(-3sin(2\u03c0((-1x + 1)/(3x + 2)) - 3) - 3) - 1) + 4 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|(-5(2cos(2\u03c0((-3x + 3)/(2x + 3))) - 2)^2 + 3(2cos(2\u03c0((-3x + 3)/(2x + 3))) - 2) + 3) - 3| + 2 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((-3(2|(-2cos(1\u03c0x))| - 2) + 2)/(-1(2|(-2cos(1\u03c0x))| - 2) + 3)) + 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) = (2(4(5(3cos(1\u03c0x - 2) - 2) - 2)^2 + 1(5(3cos(1\u03c0x - 2) - 2) - 2)) + 3)/(1(4(5(3cos(1\u03c0x - 2) - 2) - 2)^2 + 1(5(3cos(1\u03c0x - 2) - 2) - 2)) - 3) 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) = 3sin(1\u03c0((3(2|x - 2| - 2))/(-1(2|x - 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": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(3sin(3\u03c0((1x)/(-1x - 3)) + 2) + 3)^2 + 4 negative?\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(2\u03c0(3(5x - 5)^2 - 1(5x - 5) - 2)) - 3) + 3| - 3 negative?\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) = 2(3|(1cos(1\u03c0x + 2) - 3)| - 3) + 2 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) = 3cos(3\u03c0(-3|((-1x + 3)/(-2x + 1)) - 1|) + 3) negative?\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|(3(1(1cos(2\u03c0x))^2 + 4(1cos(2\u03c0x)) - 3) - 5) + 2| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4((-2(3cos(1\u03c0x - 3)) + 3)/(2(3cos(1\u03c0x - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(2sin(3\u03c0((-1x + 1)/(2x)) + 1) + 1) - 1| + 2 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) = -1(-2|(2sin(1\u03c0x - 1) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-2(1x^2 + 2x) - 5) - 1) + 1 is strictly decreasing.\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(-3(2sin(2\u03c0(2|x - 3| - 3) + 2) + 2)^2 - 5(2sin(2\u03c0(2|x - 3| - 3) + 2) + 2) + 5) + 2)/(-1(-3(2sin(2\u03c0(2|x - 3| - 3) + 2) + 2)^2 - 5(2sin(2\u03c0(2|x - 3| - 3) + 2) + 2) + 5) + 3) is strictly increasing.\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|(1sin(3\u03c0(-3((1x - 2)/(2x - 1)) - 1) - 2)) - 2| - 1 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|((2(1(1sin(3\u03c0x + 2) + 2) - 1) - 3)/(-3(1(1sin(3\u03c0x + 2) + 2) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0(-5(1x + 1)^2 + 2(1x + 1) - 4) - 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(1(2sin(1\u03c0x) - 1)^2 - 5(2sin(1\u03c0x) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1\u03c0(3|((1x - 2)/(3x - 1)) + 3|)) negative?\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) = -1(-1|(2sin(1\u03c0x) + 1) - 3| + 3) + 3 is strictly increasing.\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(2\u03c0(-4(-4x^2 + 5x + 4) + 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": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-3|(-4x^2 - 5x + 1) - 1| - 2)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(3cos(2\u03c0(2|(4x + 2) + 1| - 1) - 3) + 2)^2 + 4(3cos(2\u03c0(2|(4x + 2) + 1| - 1) - 3) + 2) + 2 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) = 1|(-3cos(3\u03c0(5(-4x^2 - 1x - 2) + 1) + 3) - 1) + 3| - 2 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(-3sin(1\u03c0(1|x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(1cos(3\u03c0x - 1) - 2) + 2)^2 + 3(1(1cos(3\u03c0x - 1) - 2) + 2) - 5 positive?\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(1|(-1sin(3\u03c0x - 1) + 1) - 1| + 3) + 4)^2 - 2(1(1|(-1sin(3\u03c0x - 1) + 1) - 1| + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(1cos(2\u03c0(-3(2|x + 2|) - 1) + 3) - 1) + 3)/(1(1cos(2\u03c0(-3(2|x + 2|) - 1) + 3) - 1)) 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) = (-1(3|(3cos(3\u03c0x + 3) + 2) + 1| + 2) - 1)/(-3(3|(3cos(3\u03c0x + 3) + 2) + 1| + 2) + 1) 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) = 3sin(1\u03c0(3|(-4(-2x^2 + 2x) + 2) + 2| + 3)) 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) = -1sin(2\u03c0((-2(-3(-2|x + 3|) - 4) + 1)/(-1(-3(-2|x + 3|) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0(-4(-1|(3x - 4) - 1| - 2)^2 - 3(-1|(3x - 4) - 1| - 2) + 4) + 1) - 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((-2(3sin(3\u03c0(3|x - 3| - 1) - 1) + 3) - 2)/(-3(3sin(3\u03c0(3|x - 3| - 1) - 1) + 3) - 2))^2 - 3((-2(3sin(3\u03c0(3|x - 3| - 1) - 1) + 3) - 2)/(-3(3sin(3\u03c0(3|x - 3| - 1) - 1) + 3) - 2)) - 1 is strictly increasing.\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) = -2|(2sin(3\u03c0(1x + 4) + 2) - 2) + 1| + 2 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) = 2|(2(3sin(2\u03c0(2x^2 + 5x + 3) - 1)) + 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": "33", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1(-2|(-1sin(1\u03c0x + 1) + 2) + 3|) + 4) - 1)/(-1(-1(-2|(-1sin(1\u03c0x + 1) + 2) + 3|) + 4)) 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) = 3((-3(1sin(1\u03c0x - 2)) - 1)/(2(1sin(1\u03c0x - 2)) + 2))^2 + 1((-3(1sin(1\u03c0x - 2)) - 1)/(2(1sin(1\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(-2(2|(5x^2 + 5) - 2| + 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) = -4((-1(3cos(3\u03c0x) - 3) + 2)/(2(3cos(3\u03c0x) - 3) - 1)) + 1 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) = -3sin(1\u03c0(1(-2|(-3x + 4) - 1|)^2 - 3(-2|(-3x + 4) - 1|) + 3) + 1) 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) = -1|(-3(2sin(2\u03c0(2x^2 + 2x + 2) - 2) - 2) - 4) + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-4(2|x| - 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": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1sin(3\u03c0((1x + 3)/(2x + 1)) - 1) + 1) 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) = 2(-3cos(3\u03c0(-3|((-3x - 3)/(-1x + 3)) - 3| - 2) - 2) + 2)^2 - 4(-3cos(3\u03c0(-3|((-3x - 3)/(-1x + 3)) - 3| - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2|(-4(2cos(1\u03c0x) - 2)^2 - 2(2cos(1\u03c0x) - 2) + 3) + 2| + 1) - 2 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(-3sin(2\u03c0(5x + 4) + 3) + 2) - 3)/(-2(-3sin(2\u03c0(5x + 4) + 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": "84", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(1\u03c0(-1(-3x^2 - 5x - 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": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2(-4(-3sin(2\u03c0x + 2) - 2) + 3)^2 + 3(-4(-3sin(2\u03c0x + 2) - 2) + 3) - 3) + 3)/(-1(-2(-4(-3sin(2\u03c0x + 2) - 2) + 3)^2 + 3(-4(-3sin(2\u03c0x + 2) - 2) + 3) - 3) + 2) 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) = 2((1(2sin(1\u03c0x + 1)))/(2(2sin(1\u03c0x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(3|(-2sin(3\u03c0x - 2) - 2)| + 1) - 3)/(2(3|(-2sin(3\u03c0x - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(3cos(2\u03c0(2|((-1x - 2)/(-2x - 3)) + 3| - 2) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3sin(2\u03c0(-1|x - 2| + 3) + 3))^2 + 4(3sin(2\u03c0(-1|x - 2| + 3) + 3)) + 1 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) = 1sin(3\u03c0(-1(5(3|x + 3| + 1)^2 + 1(3|x + 3| + 1) + 1) - 3) + 2) - 2 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) = 1|(-1sin(2\u03c0(3x + 4) + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1sin(1\u03c0(1((3x - 1)/(2x - 3))^2 + 5((3x - 1)/(2x - 3))) - 2) - 1) - 1 is strictly increasing.\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) = -3(-1|(3sin(2\u03c0x + 2)) + 1| - 3) - 2 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) = (1(2|(3sin(2\u03c0x + 1) - 3) + 1| - 3) + 1)/(-1(2|(3sin(2\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-3|(3cos(1\u03c0x - 2) - 1)|))/(1(-3|(3cos(1\u03c0x - 2) - 1)|) + 2) 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) = -1sin(2\u03c0(2(-3|((3x)/(-3x - 3)) - 1|) + 3) - 2) - 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) = -2(3cos(2\u03c0(3x - 3) - 3))^2 + 1(3cos(2\u03c0(3x - 3) - 3)) - 4 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) = -2|(4(-1sin(1\u03c0x + 3) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3cos(2\u03c0(-1x^2 - 4x + 3) + 3) + 2) + 1)/(-3(-3cos(2\u03c0(-1x^2 - 4x + 3) + 3) + 2) + 2) 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) = 4(2cos(3\u03c0(-1x^2 - 1x - 1) - 2) - 1) + 3 is strictly decreasing.\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(-3|(2(1sin(3\u03c0x + 1) - 2) - 2) - 3| - 2) - 3)/(3(-3|(2(1sin(3\u03c0x + 1) - 2) - 2) - 3| - 2) - 1) 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) = -4(-5(-1|(1cos(2\u03c0x) - 2) - 1| + 3)^2 + 1(-1|(1cos(2\u03c0x) - 2) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(1(3|x - 3| - 3)^2 - 1(3|x - 3| - 3) + 3) + 3) + 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) = -5((-2(3cos(3\u03c0x - 3) + 2) - 2)/(-1(3cos(3\u03c0x - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(2sin(2\u03c0(-2|x - 2| + 2) + 3) - 3) - 3)/(-3(2sin(2\u03c0(-2|x - 2| + 2) + 3) - 3) - 3) 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) = 1sin(1\u03c0(1|(-1x^2 + 3x + 1) - 3| + 2) - 1) + 1 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) = 3cos(2\u03c0(3|(1x^2 + 5x) - 2| - 2)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1|(1cos(2\u03c0x - 1) + 1) - 2| + 2)^2 - 5 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) = 1cos(2\u03c0((-2(-2|x - 2| + 1) - 1)/(-1(-2|x - 2| + 1)))) - 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) = -3cos(1\u03c0(-2|(-2(5x^2 - 4x - 3) - 5)| + 2) - 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) = 5(1|(5(-1sin(3\u03c0x) - 1)^2 - 2(-1sin(3\u03c0x) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(1cos(3\u03c0(4(1x - 1)^2 + 4(1x - 1)) - 3) - 1) - 1)/(-2(1cos(3\u03c0(4(1x - 1)^2 + 4(1x - 1)) - 3) - 1) + 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) = -1cos(1\u03c0(-5((3x)/(1x - 3))^2 - 4((3x)/(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": "37", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((-1(1cos(3\u03c0(-2x + 1) - 3) + 3) - 2)/(1(1cos(3\u03c0(-2x + 1) - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-1(-1(-3cos(3\u03c0x - 1) + 1) - 4)^2 + 4(-1(-3cos(3\u03c0x - 1) + 1) - 4) - 1) - 1)/(-3(-1(-1(-3cos(3\u03c0x - 1) + 1) - 4)^2 + 4(-1(-3cos(3\u03c0x - 1) + 1) - 4) - 1) - 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) = 3sin(1\u03c0(4(1x + 1)^2 - 1(1x + 1) + 3) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(2(2|x - 3| + 3)^2 - 3(2|x - 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": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(1cos(3\u03c0(1|x - 2| + 2) + 3) + 2)^2 + 2(1cos(3\u03c0(1|x - 2| + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4((-2(4(1cos(1\u03c0x + 1) + 3) - 1) - 3)/(-1(4(1cos(1\u03c0x + 1) + 3) - 1)))^2 + 5((-2(4(1cos(1\u03c0x + 1) + 3) - 1) - 3)/(-1(4(1cos(1\u03c0x + 1) + 3) - 1))) + 4 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(2cos(1\u03c0((1x + 2)/(-1x - 3)) - 3) + 2) - 1 is strictly increasing.\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((-2(4(2sin(1\u03c0x) - 2) - 4) - 3)/(-1(4(2sin(1\u03c0x) - 2) - 4) + 2))^2 + 4((-2(4(2sin(1\u03c0x) - 2) - 4) - 3)/(-1(4(2sin(1\u03c0x) - 2) - 4) + 2)) - 2 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) = -3cos(3\u03c0((1(2|x - 3| - 1) + 2)/(2(2|x - 3| - 1) - 3)) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1sin(1\u03c0(1x^2 + 5x + 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": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(2cos(1\u03c0(5(-3|x + 2| + 2) - 3) - 1))^2 - 3(2cos(1\u03c0(5(-3|x + 2| + 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": "67", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(1\u03c0(-1(-1|(4x - 2) - 2|)^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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1cos(3\u03c0(-2(3|x - 1| - 2) + 4) + 1))^2 + 2(1cos(3\u03c0(-2(3|x - 1| - 2) + 4) + 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": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3cos(1\u03c0(3x^2 - 3x - 1)) - 2) + 2 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) = 2cos(3\u03c0(-1|(-1x^2 + 5x + 3) + 3| + 3)) - 3 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) = -1sin(2\u03c0(-3|((-3x - 3)/(1x)) + 1| - 1) + 3) - 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) = -1cos(3\u03c0(3|(4x^2 + 2x) + 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": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-2cos(2\u03c0(-1|x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(-4(-3|x + 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": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((-3(-2(2cos(1\u03c0x - 3) + 2)^2 - 3(2cos(1\u03c0x - 3) + 2) - 5) + 1)/(2(-2(2cos(1\u03c0x - 3) + 2)^2 - 3(2cos(1\u03c0x - 3) + 2) - 5) + 3)) + 1 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) = -1(5(3sin(3\u03c0(2|x + 3| + 2) + 3) - 1) + 4)^2 + 4(5(3sin(3\u03c0(2|x + 3| + 2) + 3) - 1) + 4) + 1 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(1|(-1(-2cos(2\u03c0x) + 3)^2 - 5(-2cos(2\u03c0x) + 3) - 3)| - 3) - 2)/(-2(1|(-1(-2cos(2\u03c0x) + 3)^2 - 5(-2cos(2\u03c0x) + 3) - 3)| - 3) - 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(3cos(3\u03c0(-4x + 5) + 1) - 1)^2 - 5(3cos(3\u03c0(-4x + 5) + 1) - 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2sin(1\u03c0((-1x - 1)/(1x)) + 1) + 1)^2 - 3(-2sin(1\u03c0((-1x - 1)/(1x)) + 1) + 1) - 4 is strictly increasing.\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|(-1(2sin(3\u03c0x + 3) + 1)^2 - 5(2sin(3\u03c0x + 3) + 1) + 1) - 3| - 2 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) = (2(-1(1sin(3\u03c0x - 2) - 3)^2 + 4(1sin(3\u03c0x - 2) - 3) + 1) - 2)/(2(-1(1sin(3\u03c0x - 2) - 3)^2 + 4(1sin(3\u03c0x - 2) - 3) + 1) + 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) = 1(-2(1sin(1\u03c0x + 1) + 3) - 2)^2 + 4(-2(1sin(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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(5(2sin(1\u03c0(1|x - 2| - 1)) - 2) - 4))/(-1(5(2sin(1\u03c0(1|x - 2| - 1)) - 2) - 4) + 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) = (1(3sin(1\u03c0(3|x + 1| - 1) + 2) - 2) - 2)/(3(3sin(1\u03c0(3|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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0(-3((3(-3x - 2) + 3)/(-1(-3x - 2) + 3))^2 - 2((3(-3x - 2) + 3)/(-1(-3x - 2) + 3)) - 3) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(-3|(-2(-3x - 2)^2 - 4(-3x - 2) - 3) + 3| + 3) + 3) + 1 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(-3|((-1(-1sin(3\u03c0x - 2) + 3) + 3)/(-3(-1sin(3\u03c0x - 2) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0((-3(-4x^2 + 3x - 1) - 1)/(-3(-4x^2 + 3x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3\u03c0(4(-1x^2 + 4x + 3) - 5) - 3) - 2 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) = 1sin(2\u03c0((2(-1x - 3) - 2)/(3(-1x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-3|(3sin(2\u03c0x) - 3) + 2| + 1) + 3)/(3(-3|(3sin(2\u03c0x) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-1(-2sin(2\u03c0x + 1) - 3)^2 - 1(-2sin(2\u03c0x + 1) - 3) - 5) + 2)/(-2(-1(-2sin(2\u03c0x + 1) - 3)^2 - 1(-2sin(2\u03c0x + 1) - 3) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((2(2cos(3\u03c0x - 3)) - 2)/(-1(2cos(3\u03c0x - 3)) + 3))^2 + 4((2(2cos(3\u03c0x - 3)) - 2)/(-1(2cos(3\u03c0x - 3)) + 3)) - 3 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) = -3sin(1\u03c0(-3|(1x^2 - 4x + 3) + 2| + 2) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(1cos(3\u03c0(-4x^2 + 5x + 1) - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3cos(2\u03c0((2x)/(2x - 1)) + 3) - 1)^2 + 1(-3cos(2\u03c0((2x)/(2x - 1)) + 3) - 1) - 4 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(-1sin(1\u03c0(1|(-1x^2 + 5x - 1) + 2| + 1) - 2) + 3) - 1)/(-1(-1sin(1\u03c0(1|(-1x^2 + 5x - 1) + 2| + 1) - 2) + 3) - 1) 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) = 2|(2cos(2\u03c0(-3x^2 - 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": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(3(2(3cos(1\u03c0x - 3) - 3)^2 + 2(3cos(1\u03c0x - 3) - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(1(2|(3sin(3\u03c0x - 2) - 3) + 1| + 1)^2 + 3(2|(3sin(3\u03c0x - 2) - 3) + 1| + 1) - 5) + 2)/(1(1(2|(3sin(3\u03c0x - 2) - 3) + 1| + 1)^2 + 3(2|(3sin(3\u03c0x - 2) - 3) + 1| + 1) - 5)) 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) = 1sin(1\u03c0(-1|(5((3x + 3)/(-3x - 2))^2 - 2((3x + 3)/(-3x - 2)) + 3) + 3|) - 3) - 1 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) = -3|((1(-3cos(1\u03c0(2x + 3) + 2) - 2) + 1)/(-1(-3cos(1\u03c0(2x + 3) + 2) - 2) - 2)) - 3| is strictly decreasing.\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(5(-2sin(1\u03c0(-2|x + 1| + 2) - 1) + 3)^2 + 2(-2sin(1\u03c0(-2|x + 1| + 2) - 1) + 3) - 5) - 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) = 3(3|(2cos(2\u03c0(-4x + 3) - 1) - 1)| - 3)^2 + 2(3|(2cos(2\u03c0(-4x + 3) - 1) - 1)| - 3) is strictly increasing.\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(1\u03c0(2|((3x + 2)/(-2x + 1)) - 3| - 2) + 2) - 2 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) = (-2(5(2sin(2\u03c0(-1x^2 + 4x) + 2)) - 4) - 3)/(-1(5(2sin(2\u03c0(-1x^2 + 4x) + 2)) - 4) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(1\u03c0(-4(2|x| - 1)^2 - 4(2|x| - 1) - 5) - 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) = -2cos(2\u03c0(3((-2x + 3)/(2x + 1))^2 - 2((-2x + 3)/(2x + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-1|(3(1x - 5)^2 + 3(1x - 5) + 1) + 3| - 2) - 2) + 1 is strictly increasing.\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) = (-1(2|(3(-1cos(1\u03c0x - 1)) + 4) - 3| + 1) + 2)/(1(2|(3(-1cos(1\u03c0x - 1)) + 4) - 3| + 1)) 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) = -1(-1|(2cos(1\u03c0x - 2) - 1) - 1| + 1) - 5 is strictly increasing.\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(-1sin(2\u03c0x - 3) - 1)^2 + 4(-1sin(2\u03c0x - 3) - 1) + 4) - 1)/(-1(-2(-1sin(2\u03c0x - 3) - 1)^2 + 4(-1sin(2\u03c0x - 3) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4((3(-1|(-2sin(1\u03c0x + 1))| - 2))/(-1(-1|(-2sin(1\u03c0x + 1))| - 2) + 2)) - 5 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((-1(1(2|x + 3| + 1) - 2) - 3)/(-3(1(2|x + 3| + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(3sin(1\u03c0(-3x + 4)) + 1) + 3)/(-2(3sin(1\u03c0(-3x + 4)) + 1) - 3) 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|(3sin(3\u03c0x - 2) + 1) + 3| + 2) + 3)/(-1(1|(3sin(3\u03c0x - 2) + 1) + 3| + 2) - 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) = (-1(2(2sin(3\u03c0(-5x^2 + 3x + 3) - 2) - 2)) + 2)/(1(2(2sin(3\u03c0(-5x^2 + 3x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2(-1cos(3\u03c0(2x^2) - 3) + 1) + 1) + 3)/(-2(-2(-1cos(3\u03c0(2x^2) - 3) + 1) + 1)) is strictly decreasing.\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((-1(-1sin(3\u03c0x + 2) + 1) + 3)/(1(-1sin(3\u03c0x + 2) + 1) + 3))^2 - 4((-1(-1sin(3\u03c0x + 2) + 1) + 3)/(1(-1sin(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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-1cos(2\u03c0x + 1) + 1)^2 - 2(-1cos(2\u03c0x + 1) + 1) - 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) = 1(5(3sin(3\u03c0(-2|x| - 2)) + 1) - 1)^2 - 4(5(3sin(3\u03c0(-2|x| - 2)) + 1) - 1) + 1 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) = -1((3(3sin(1\u03c0x - 2) + 3))/(-2(3sin(1\u03c0x - 2) + 3) + 3)) - 1 negative?\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((-2x - 3)/(-1x - 2)) + 2) - 2)^2 - 3(-3cos(3\u03c0((-2x - 3)/(-1x - 2)) + 2) - 2) + 5) + 2| - 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|(-2cos(2\u03c0(2x - 2) - 1)) - 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) = 2sin(1\u03c0(3|(-4x + 3) - 3| - 2) + 3) - 3 is strictly decreasing.\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) = -2sin(1\u03c0((2(-3|x - 1| + 1) + 1)/(3(-3|x - 1| + 1) + 3)) + 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) = -1|(2cos(3\u03c0(4(1x^2 + 2x - 4) + 1) + 3) + 1) - 1| 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) = -4((-1(2sin(3\u03c0x) + 3) + 3)/(-3(2sin(3\u03c0x) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(1\u03c0(3|(-1x^2 - 5x + 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": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1cos(1\u03c0(1x + 2) + 2) - 2) - 1)/(2(-1cos(1\u03c0(1x + 2) + 2) - 2) + 2) 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) = 5(3|(-1cos(1\u03c0(2x - 5)) + 2) - 2| - 2)^2 - 3(3|(-1cos(1\u03c0(2x - 5)) + 2) - 2| - 2) + 2 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(1(3sin(3\u03c0(-1|x + 2| + 1) + 1) - 2) - 1)^2 + 4(1(3sin(3\u03c0(-1|x + 2| + 1) + 1) - 2) - 1) - 1 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) = -2|(5(-2cos(3\u03c0x) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1(-2sin(3\u03c0x - 2) - 2) + 4) - 2| + 1 negative?\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(1cos(3\u03c0((-2x - 1)/(-1x)) - 3) - 1)^2 - 4(1cos(3\u03c0((-2x - 1)/(-1x)) - 3) - 1) - 5 negative?\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(2(-3(-3cos(1\u03c0x - 1) - 1)^2 - 1(-3cos(1\u03c0x - 1) - 1) + 5) - 1) - 2)/(-2(2(-3(-3cos(1\u03c0x - 1) - 1)^2 - 1(-3cos(1\u03c0x - 1) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0(-5(-1x^2 + 2x + 4) - 4) + 3) + 1 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) = -4(-1(1cos(2\u03c0x - 3) - 1))^2 + 2(-1(1cos(2\u03c0x - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2|(2cos(3\u03c0x - 2) + 1)| - 2)^2 + 3(2|(2cos(3\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": "57", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0((-3(1x^2 + 2x + 4) - 1)/(-1(1x^2 + 2x + 4) + 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": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-3cos(2\u03c0(4((-2x + 1)/(-1x - 1))^2 - 5((-2x + 1)/(-1x - 1)) + 4) + 3) - 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-3sin(1\u03c0(-1x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((-3(-3(1cos(1\u03c0x + 3) - 3)^2 + 1(1cos(1\u03c0x + 3) - 3)) - 2)/(-1(-3(1cos(1\u03c0x + 3) - 3)^2 + 1(1cos(1\u03c0x + 3) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(4(-2sin(2\u03c0x - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(1|(5(-1cos(1\u03c0x - 2) - 2) + 3) - 1| + 1) + 3)/(-2(1|(5(-1cos(1\u03c0x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|((2(-3(3sin(2\u03c0x - 1)) - 2))/(-1(-3(3sin(2\u03c0x - 1)) - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(2cos(2\u03c0(2x^2 - 5x - 5) + 1) + 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": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2sin(3\u03c0(-2|x + 2| + 2)) + 2) + 4 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) = (-3(1sin(3\u03c0(1|x - 2| + 3) + 1) + 2) - 2)/(-2(1sin(3\u03c0(1|x - 2| + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2(1sin(3\u03c0x + 1) - 1) - 1)^2 - 2(2(1sin(3\u03c0x + 1) - 1) - 1) + 1 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) = -3cos(2\u03c0((-1(-2x + 1) + 2)/(1(-2x + 1) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(1|(-5x^2 + 5x - 5) - 1| + 1) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(1(-3sin(1\u03c0x - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1(-2sin(1\u03c0x - 2) - 3) + 4)^2 + 3(1(-2sin(1\u03c0x - 2) - 3) + 4) + 1 negative?\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) = -1cos(1\u03c0(-2|(-5(-3x^2 + 3x) - 1) - 1| - 2) - 2) + 2 negative?\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|((-3(-2sin(2\u03c0x + 2) + 2))/(-3(-2sin(2\u03c0x + 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) = 5(4((3(-3sin(2\u03c0x + 2)) - 1)/(2(-3sin(2\u03c0x + 2)) + 2)) + 2)^2 + 3(4((3(-3sin(2\u03c0x + 2)) - 1)/(2(-3sin(2\u03c0x + 2)) + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-5(3(2cos(1\u03c0x - 1) - 1)^2 + 2(2cos(1\u03c0x - 1) - 1) + 3) - 4) + 1| - 2 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) = -3cos(3\u03c0(-2|(-3x + 1) - 3| - 1) - 1) - 3 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) = 3(-2|(-3cos(1\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2(-1sin(3\u03c0x - 1) - 1) + 5)^2 - 1(-2(-1sin(3\u03c0x - 1) - 1) + 5) + 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) = -3(3|(1cos(2\u03c0x + 3) - 3) + 1| - 3)^2 - 3(3|(1cos(2\u03c0x + 3) - 3) + 1| - 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) = (-3(-5(-2sin(2\u03c0x - 2) - 1)^2 - 4(-2sin(2\u03c0x - 2) - 1) + 4) - 3)/(3(-5(-2sin(2\u03c0x - 2) - 1)^2 - 4(-2sin(2\u03c0x - 2) - 1) + 4) - 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) = 1(-1(-2cos(3\u03c0x - 3) + 2) - 1)^2 - 2(-1(-2cos(3\u03c0x - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1sin(1\u03c0(-4(2|x + 3| - 3) + 5) + 3) + 2) - 1)/(3(-1sin(1\u03c0(-4(2|x + 3| - 3) + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((-1(-4(-3cos(1\u03c0x + 3) - 3)^2 - 5(-3cos(1\u03c0x + 3) - 3)) + 1)/(3(-4(-3cos(1\u03c0x + 3) - 3)^2 - 5(-3cos(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(1sin(2\u03c0(-3|x - 2| + 2) - 3))^2 + 1(1sin(2\u03c0(-3|x - 2| + 2) - 3)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3(-1sin(1\u03c0x + 3) + 3)^2 - 2(-1sin(1\u03c0x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(5(1cos(2\u03c0x + 3) + 1)^2 + 2(1cos(2\u03c0x + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((-2(-2cos(1\u03c0(-1|x + 2| - 2) - 3) + 3) + 1)/(-3(-2cos(1\u03c0(-1|x + 2| - 2) - 3) + 3) + 2)) + 5 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) = 3cos(3\u03c0(-2|(-3((-3x - 1)/(-3x + 4)) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2cos(3\u03c0(1x^2 - 5x) + 2) - 1) + 4 positive?\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) = (2(-1cos(3\u03c0(1x + 5) - 3) - 2) + 1)/(1(-1cos(3\u03c0(1x + 5) - 3) - 2)) 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) = 3|(4(1sin(1\u03c0x + 2) + 2)^2 + 4(1sin(1\u03c0x + 2) + 2) - 3) + 3| - 2 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) = -4(-1sin(1\u03c0(-1(3|x + 1| + 3)^2 - 5(3|x + 1| + 3) + 4) + 1) + 1) - 3 negative?\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(-4(-2(3cos(3\u03c0x - 1) - 2) - 1)^2 + 2(-2(3cos(3\u03c0x - 1) - 2) - 1) - 4) + 2)/(-1(-4(-2(3cos(3\u03c0x - 1) - 2) - 1)^2 + 2(-2(3cos(3\u03c0x - 1) - 2) - 1) - 4) + 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) = -5(2(2sin(2\u03c0x - 2) + 2)^2 + 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) = 3sin(3\u03c0(-2|(-3(1x^2 + 4x - 2) - 2) - 2| - 1) + 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(1cos(1\u03c0(-5x^2 - 4) - 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": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((3(-2sin(1\u03c0(-3x + 3) - 3) - 3) - 2)/(-3(-2sin(1\u03c0(-3x + 3) - 3) - 3) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(1\u03c0(-5(-2|x + 3| + 2) - 2) + 3) - 1 negative?\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) = 1(2|(-1sin(1\u03c0x) + 1) - 1| - 1) 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) = 3sin(3\u03c0(2(-1|((2x + 3)/(-2x + 3)) + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(3sin(1\u03c0(-2|(-1x^2 + 4x - 3) - 1| - 1) - 3) + 3) - 2)/(-1(3sin(1\u03c0(-2|(-1x^2 + 4x - 3) - 1| - 1) - 3) + 3) + 3) 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) = 4((-1(1(-1sin(1\u03c0x + 1)) - 2) + 3)/(-1(1(-1sin(1\u03c0x + 1)) - 2) - 3))^2 - 1((-1(1(-1sin(1\u03c0x + 1)) - 2) + 3)/(-1(1(-1sin(1\u03c0x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(1\u03c0(-3|((-3x + 1)/(-1x)) - 2| + 3) + 1) + 3 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(1\u03c0((3(2|(1x^2 - 2x) - 1| - 1) - 2)/(1(2|(1x^2 - 2x) - 1| - 1))) + 3) - 2 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) = 1(-3|(-3sin(2\u03c0x + 2) + 2) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-5(-2cos(3\u03c0(-1x - 2)) - 1)^2 - 4(-2cos(3\u03c0(-1x - 2)) - 1) + 4)| is strictly increasing.\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) = 2cos(2\u03c0(-2((1(3x - 4) - 2)/(-3(3x - 4) - 3))^2 - 4((1(3x - 4) - 2)/(-3(3x - 4) - 3)) - 4) - 2) + 3 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) = 4(2cos(3\u03c0((1x)/(-1x - 3)) + 1)) + 4 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) = 1(-5(-3sin(2\u03c0x))^2 - 1(-3sin(2\u03c0x)) + 4) - 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) = (-3(-1(1cos(1\u03c0(1x - 2) - 3) - 1)^2 - 4(1cos(1\u03c0(1x - 2) - 3) - 1) - 4))/(-2(-1(1cos(1\u03c0(1x - 2) - 3) - 1)^2 - 4(1cos(1\u03c0(1x - 2) - 3) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1sin(1\u03c0((-1(-2x + 1) + 1)/(1(-2x + 1) + 1)) - 2) + 2) + 3| + 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) = 4((-2(1sin(3\u03c0(-1|x + 3| - 2) - 1) + 1) + 3)/(1(1sin(3\u03c0(-1|x + 3| - 2) - 1) + 1) - 3)) + 4 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) = 2cos(2\u03c0(-1|((3x + 1)/(2x - 1)) - 3| - 2) - 3) - 2 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) = 4(-5(-2|(-2cos(3\u03c0x - 3) + 1) + 1| + 2)^2 - 1(-2|(-2cos(3\u03c0x - 3) + 1) + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-1(-1sin(1\u03c0(1x + 1) + 2) - 2)^2 - 3(-1sin(1\u03c0(1x + 1) + 2) - 2)) + 3)/(2(-1(-1sin(1\u03c0(1x + 1) + 2) - 2)^2 - 3(-1sin(1\u03c0(1x + 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) = 2sin(3\u03c0(2|(2x - 2) - 1| - 3) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-3(-1cos(3\u03c0(3x + 1) - 2) - 2)^2 - 2(-1cos(3\u03c0(3x + 1) - 2) - 2) - 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": "68", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0((-1(-3|x + 2| - 2) + 3)/(2(-3|x + 2| - 2) + 2)) + 2) + 2 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) = -1sin(3\u03c0(-1|(-1((-2x - 3)/(-1x + 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": "For the function f(x) = 2|((1(-3(-1cos(1\u03c0x - 2) + 1)^2 - 3(-1cos(1\u03c0x - 2) + 1)) - 1)/(3(-3(-1cos(1\u03c0x - 2) + 1)^2 - 3(-1cos(1\u03c0x - 2) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(1|(-3cos(2\u03c0x - 1) + 1) - 3|))/(2(1|(-3cos(2\u03c0x - 1) + 1) - 3|) + 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) = 2((2(3sin(1\u03c0(-2|x - 2| + 3) - 2) - 2) - 1)/(-1(3sin(1\u03c0(-2|x - 2| + 3) - 2) - 2)))^2 + 4((2(3sin(1\u03c0(-2|x - 2| + 3) - 2) - 2) - 1)/(-1(3sin(1\u03c0(-2|x - 2| + 3) - 2) - 2))) - 5 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) = -2|(-2cos(2\u03c0(-1x + 5) - 2)) + 3| + 2 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) = -1(1sin(1\u03c0(-1x^2 - 4x - 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": "For the function f(x) = 2|(-1cos(1\u03c0(2(-1x - 3)^2 - 1(-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": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(3\u03c0(-1(1(-2|x + 2| - 3)^2 - 5(-2|x + 2| - 3) + 4) - 1)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((-1(-1sin(1\u03c0x) - 2) + 2)/(-3(-1sin(1\u03c0x) - 2) - 1)) + 2| - 2 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) = 2sin(2\u03c0(5(1x^2 - 4x - 4)) + 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) = 2sin(3\u03c0((-2(-4x + 1) + 3)/(1(-4x + 1))) - 2) + 2 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) = 3((2(-3cos(3\u03c0(-3x + 2) - 3) - 1) - 2)/(1(-3cos(3\u03c0(-3x + 2) - 3) - 1)))^2 + 4((2(-3cos(3\u03c0(-3x + 2) - 3) - 1) - 2)/(1(-3cos(3\u03c0(-3x + 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": "63", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2cos(2\u03c0(-5(-1|x - 1| + 2)^2 + 1(-1|x - 1| + 2) + 5) + 3)) - 1)/(2(2cos(2\u03c0(-5(-1|x - 1| + 2)^2 + 1(-1|x - 1| + 2) + 5) + 3)) + 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) = -2|(3((3(2sin(1\u03c0x + 3) + 3) - 1)/(2(2sin(1\u03c0x + 3) + 3))) + 3) - 2| + 1 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) = -4(-1((-1(3sin(1\u03c0x - 2) - 3))/(-3(3sin(1\u03c0x - 2) - 3) - 2))^2 - 5((-1(3sin(1\u03c0x - 2) - 3))/(-3(3sin(1\u03c0x - 2) - 3) - 2)) - 1) - 5 positive?\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|((-1(-1(-2cos(1\u03c0x - 2)) - 1))/(1(-1(-2cos(1\u03c0x - 2)) - 1) + 3)) + 3| - 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(2|(-2cos(2\u03c0x - 1) + 2) + 1| - 1)^2 + 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) = 2cos(3\u03c0(-2(3x^2 + 3x + 4) + 2) - 1) + 1 is strictly decreasing.\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) = 3((3(-1cos(3\u03c0(1x^2 - 3x - 1) - 3) - 3) + 3)/(1(-1cos(3\u03c0(1x^2 - 3x - 1) - 3) - 3) + 2)) + 3 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) = (3(5(-3sin(1\u03c0x) + 1) + 3) + 3)/(-3(5(-3sin(1\u03c0x) + 1) + 3) - 2) 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) = 3cos(3\u03c0(4(1((2x + 2)/(-1x + 3))^2 - 1((2x + 2)/(-1x + 3)) + 1) - 3) - 1) - 1 positive?\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) = 2(-3sin(3\u03c0(-1(-1|x + 2| - 2) - 3) - 2) - 3)^2 + 4(-3sin(3\u03c0(-1(-1|x + 2| - 2) - 3) - 2) - 3) - 2 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) = 4(5(-2cos(3\u03c0x - 3) - 2) - 2)^2 - 5(5(-2cos(3\u03c0x - 3) - 2) - 2) - 1 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) = 1(3sin(1\u03c0((-3x)/(2x + 3)) - 3) + 2)^2 - 2(3sin(1\u03c0((-3x)/(2x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2|(-1cos(3\u03c0((-3x)/(-3x - 3)) + 3)) - 1| + 3) - 5 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) = 2((-2(-3sin(2\u03c0(1|x - 1| + 2) - 2) - 3) - 1)/(-1(-3sin(2\u03c0(1|x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(1cos(3\u03c0(5(-3|x|)) + 3) - 3) + 3)/(-3(1cos(3\u03c0(5(-3|x|)) + 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) = 2cos(3\u03c0(1(-1x^2 - 4x - 4) + 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": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2|(2(1sin(1\u03c0x) - 2)^2 + 4(1sin(1\u03c0x) - 2) - 3) + 1| - 2) + 5 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) = -2|((-1(-2sin(3\u03c0x - 2)) + 3)/(3(-2sin(3\u03c0x - 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(-3sin(3\u03c0(-1x + 5) + 2) - 1))/(1(-3sin(3\u03c0(-1x + 5) + 2) - 1) + 3) 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) = -2sin(2\u03c0(-2(-3|x + 2| + 3) + 1) + 2) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1(1sin(2\u03c0(2|x + 3| + 1) - 3) - 2)^2 - 1(1sin(2\u03c0(2|x + 3| + 1) - 3) - 2) + 1) + 5 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) = (-3(-4(-1cos(2\u03c0x - 2) - 2) - 1) + 2)/(-1(-4(-1cos(2\u03c0x - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1sin(1\u03c0(1x + 2) + 1)) - 3)/(-1(-1sin(1\u03c0(1x + 2) + 1))) 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) = -1sin(3\u03c0(-3|(5(4x^2 - 2x + 5) - 2)| + 1) + 3) negative?\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|(2(-4(3cos(1\u03c0x - 3) + 2)^2 - 2(3cos(1\u03c0x - 3) + 2) - 1) - 4) - 1| + 1 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(-1cos(1\u03c0(-3|x + 3| + 3) + 1) + 3) - 1)/(3(-1cos(1\u03c0(-3|x + 3| + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((-3(5(-1cos(1\u03c0x + 1) - 3) + 2) - 1)/(-1(5(-1cos(1\u03c0x + 1) - 3) + 2) - 2))^2 - 3((-3(5(-1cos(1\u03c0x + 1) - 3) + 2) - 1)/(-1(5(-1cos(1\u03c0x + 1) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(1(2(-1cos(2\u03c0x + 1) + 3) + 1)^2 + 4(2(-1cos(2\u03c0x + 1) + 3) + 1) - 2) + 1)/(-1(1(2(-1cos(2\u03c0x + 1) + 3) + 1)^2 + 4(2(-1cos(2\u03c0x + 1) + 3) + 1) - 2) + 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(-1(2|(-3cos(2\u03c0x - 1) - 2) + 2|) - 2)^2 + 3(-1(2|(-3cos(2\u03c0x - 1) - 2) + 2|) - 2) + 3 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) = -1sin(1\u03c0(1|(1(-5x^2 - 4x - 4) + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1cos(2\u03c0(3x^2 - 4x + 3) + 1)) + 2 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) = -2|(-2(2(2cos(3\u03c0x + 2) + 1)^2 - 2(2cos(3\u03c0x + 2) + 1) + 5) + 3)| - 1 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) = -3sin(2\u03c0(2|(2(-2x - 2)^2 + 5(-2x - 2) + 1) - 1|) + 3) + 3 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) = -2(3sin(1\u03c0(5(-1|x + 3|)) + 1) + 1)^2 + 3(3sin(1\u03c0(5(-1|x + 3|)) + 1) + 1) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "92", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2|(-4(-2cos(1\u03c0x + 2) - 1) - 1)| + 2)^2 - 3(-2|(-4(-2cos(1\u03c0x + 2) - 1) - 1)| + 2) - 2 is strictly decreasing.\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|(1(1sin(2\u03c0x - 1) - 1) - 1) + 1| + 3 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((3(1x - 5) - 3)/(-1(1x - 5))) + 2) - 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) = -1cos(2\u03c0(-1|((1x + 3)/(-3x + 4)) + 2| - 1) + 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-2(-2cos(1\u03c0x + 1) - 1) + 3)^2 - 3(-2(-2cos(1\u03c0x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-1sin(2\u03c0(3x^2 + 5x + 1) + 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": "45", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-2(5(2cos(3\u03c0x - 2)) + 5)^2 + 3(5(2cos(3\u03c0x - 2)) + 5) + 4) + 3)/(-1(-2(5(2cos(3\u03c0x - 2)) + 5)^2 + 3(5(2cos(3\u03c0x - 2)) + 5) + 4) + 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) = 5(-1sin(3\u03c0(-2x^2 - 3x) + 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": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((3(-1cos(2\u03c0(-1|x + 1| + 1) + 3)) - 3)/(-1(-1cos(2\u03c0(-1|x + 1| + 1) + 3)) - 2)) + 1 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) = 5(3(-1sin(1\u03c0x + 2))^2 - 5(-1sin(1\u03c0x + 2)) + 2) - 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) = -3|((2(-3sin(2\u03c0x + 2) + 2) - 1)/(3(-3sin(2\u03c0x + 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) = (2(-1|(-1sin(3\u03c0(3x - 5) - 2) + 2) - 3| - 3) + 2)/(-2(-1|(-1sin(3\u03c0(3x - 5) - 2) + 2) - 3| - 3)) positive?\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) = -2(2cos(1\u03c0(1|x + 3| + 2) - 3) - 2)^2 - 5(2cos(1\u03c0(1|x + 3| + 2) - 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) = -4((-2(3cos(1\u03c0(1x + 3) + 1) - 3) + 1)/(-1(3cos(1\u03c0(1x + 3) + 1) - 3) - 1))^2 - 1((-2(3cos(1\u03c0(1x + 3) + 1) - 3) + 1)/(-1(3cos(1\u03c0(1x + 3) + 1) - 3) - 1)) 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) = (3(-3(2cos(1\u03c0x - 3) + 2)^2 - 5(2cos(1\u03c0x - 3) + 2) - 4) - 2)/(1(-3(2cos(1\u03c0x - 3) + 2)^2 - 5(2cos(1\u03c0x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(1(-5(-1sin(2\u03c0x - 2))^2 + 4(-1sin(2\u03c0x - 2))) - 5) - 3| + 1 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) = -2|((-1(2sin(3\u03c0(-5x^2 - 2x - 2) - 3)) + 1)/(-2(2sin(3\u03c0(-5x^2 - 2x - 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": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1sin(1\u03c0(-2|x + 3| + 3) - 3) - 1) - 1)/(1(-1sin(1\u03c0(-2|x + 3| + 3) - 3) - 1)) 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) = -1(1((3(-2sin(1\u03c0x + 1) - 1) - 3)/(-2(-2sin(1\u03c0x + 1) - 1) - 2)) - 4)^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": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2|(-2cos(3\u03c0(3x + 2) - 3) - 3) + 1| - 3)^2 + 2(2|(-2cos(3\u03c0(3x + 2) - 3) - 3) + 1| - 3) - 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(2|(-1cos(3\u03c0(-4x + 2) - 3)) - 3|) + 2)/(1(2|(-1cos(3\u03c0(-4x + 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": "82", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(3((-3(1|x - 1| - 3) - 1)/(3(1|x - 1| - 3) - 1))) - 2) + 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) = (-1(5(-1cos(2\u03c0x - 1))^2 + 1(-1cos(2\u03c0x - 1)) + 3) - 2)/(3(5(-1cos(2\u03c0x - 1))^2 + 1(-1cos(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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(3(-1sin(2\u03c0(4x^2 - 2x - 1) - 3) + 2) + 2) + 1| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3cos(2\u03c0((-2(1x + 5) + 2)/(-2(1x + 5) + 3)) + 3) + 3)^2 - 3(3cos(2\u03c0((-2(1x + 5) + 2)/(-2(1x + 5) + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-1sin(3\u03c0((2x - 3)/(-1x - 2)) - 1) - 2) + 5 is strictly increasing.\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) = 3sin(3\u03c0((2(2|(3x^2 + 4x - 5) + 1| + 2) + 3)/(-2(2|(3x^2 + 4x - 5) + 1| + 2) + 3)) + 1) - 1 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) = 1sin(3\u03c0(-5((-3x)/(2x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0((1(2(-1|x - 2|)^2 - 2(-1|x - 2|) - 2) + 2)/(-3(2(-1|x - 2|)^2 - 2(-1|x - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0((-1(-1|x + 2| + 2) - 2)/(1(-1|x + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(2|(1cos(1\u03c0x + 3) + 1) - 3| - 1) + 1)/(-2(2|(1cos(1\u03c0x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1sin(2\u03c0((1(1|x + 1| + 1) + 3)/(-2(1|x + 1| + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(2\u03c0(-5((2x - 3)/(1x - 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": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2sin(3\u03c0(4(-2|x| + 1)^2 + 2(-2|x| + 1) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3(-1cos(3\u03c0(2x^2 - 3x - 1) - 3) - 1) + 1) - 1)/(-1(-3(-1cos(3\u03c0(2x^2 - 3x - 1) - 3) - 1) + 1) - 2) 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(-1|(-2cos(1\u03c0x + 3) - 1)| - 1)^2 + 4(-1|(-2cos(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(5(3cos(3\u03c0x - 1) + 1)^2 - 2(3cos(3\u03c0x - 1) + 1) + 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1sin(2\u03c0(3|(-3x + 3) - 3| - 2) - 1) + 1) - 3)/(-2(1sin(2\u03c0(3|(-3x + 3) - 3| - 2) - 1) + 1) - 1) 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(-3sin(2\u03c0(2(3x^2 + 4x + 2) + 4) - 2) + 3))/(3(-3sin(2\u03c0(2(3x^2 + 4x + 2) + 4) - 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": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-5(-1cos(3\u03c0x + 3) + 1)^2 + 5(-1cos(3\u03c0x + 3) + 1) + 3) - 1 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) = 3cos(2\u03c0(3|(-3x + 5) - 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-3(2cos(1\u03c0(1|x + 1| + 1)))^2 - 2(2cos(1\u03c0(1|x + 1| + 1)))) + 4 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|(-1cos(1\u03c0(-4x - 4) - 2) - 3)| 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) = (-2(2sin(1\u03c0(2(-2x^2 + 5x - 3)) - 2) + 1) + 2)/(-2(2sin(1\u03c0(2(-2x^2 + 5x - 3)) - 2) + 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2sin(2\u03c0(-3(-3x - 3)^2 - 2(-3x - 3) + 3) + 1)) + 3| - 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) = -3|(2((2(-1cos(2\u03c0x - 1) - 2) - 1)/(3(-1cos(2\u03c0x - 1) - 2) - 1))^2 - 2((2(-1cos(2\u03c0x - 1) - 2) - 1)/(3(-1cos(2\u03c0x - 1) - 2) - 1)) + 5)| + 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) = 1e^(-1(-5((3(-1|x - 3| - 1) + 3)/(1(-1|x - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-2(1|(1sin(3\u03c0x + 1) + 3) + 1|)^2 + 1(1|(1sin(3\u03c0x + 1) + 3) + 1|) - 4) + 2)/(2(-2(1|(1sin(3\u03c0x + 1) + 3) + 1|)^2 + 1(1|(1sin(3\u03c0x + 1) + 3) + 1|) - 4) - 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) = (-1(-3cos(1\u03c0(-3(5x)^2 + 3(5x) - 3) + 2) - 2) - 2)/(2(-3cos(1\u03c0(-3(5x)^2 + 3(5x) - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(3(3cos(2\u03c0x + 1) - 3)^2 - 5(3cos(2\u03c0x + 1) - 3) + 4) + 1)/(2(3(3cos(2\u03c0x + 1) - 3)^2 - 5(3cos(2\u03c0x + 1) - 3) + 4) - 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|(-1sin(1\u03c0(-5x^2 - 3x - 2) - 2) - 3) - 3| + 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) = 2sin(2\u03c0(3|(-2x + 1) + 3| + 3) + 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) = -1|(-1cos(1\u03c0(-5x - 2) - 2) + 3) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0(-1|(5x^2 - 5x + 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": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-2|(2x)| + 3) - 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) = 5(-1cos(3\u03c0(-2|((-3x + 3)/(3x)) - 3| + 2)) + 3) - 1 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) = -1sin(2\u03c0(1|(-5x + 4) - 1| + 3) + 1) + 1 is strictly decreasing.\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) = -2sin(2\u03c0(1(-3x + 3)^2 - 2(-3x + 3) - 4) + 2) 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) = 3(-1(-2sin(3\u03c0x + 2) - 2) - 1)^2 + 1(-1(-2sin(3\u03c0x + 2) - 2) - 1) - 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) = (-3(-5(2sin(2\u03c0(4x^2 - 3x - 1) + 2) + 3) - 5) - 2)/(-3(-5(2sin(2\u03c0(4x^2 - 3x - 1) + 2) + 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": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-2(-1cos(1\u03c0x - 3) + 3)^2 + 1(-1cos(1\u03c0x - 3) + 3) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((-1(-2sin(3\u03c0x + 1) - 1) - 2)/(-1(-2sin(3\u03c0x + 1) - 1) - 3)) - 2 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|((-3(2sin(2\u03c0(-1x^2 + 2x + 3) - 1) - 1) + 1)/(2(2sin(2\u03c0(-1x^2 + 2x + 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": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|(-1sin(3\u03c0x + 1)) + 3|) - 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) = -1|(-3(-3cos(3\u03c0(4x^2 - 1x - 1) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(2(3|x - 2| - 3)^2 + 2) - 2) - 2 positive?\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(2\u03c0(5((-1x + 2)/(3x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1cos(3\u03c0((1(-3|x - 3| - 1) + 1)/(1(-3|x - 3| - 1) + 3)) + 2) + 2)^2 + 3(-1cos(3\u03c0((1(-3|x - 3| - 1) + 1)/(1(-3|x - 3| - 1) + 3)) + 2) + 2) - 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) = (-1(-2cos(3\u03c0(-3|(3x^2 + 3x + 4)| + 3) + 2) + 1) + 3)/(-3(-2cos(3\u03c0(-3|(3x^2 + 3x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0(3(1|((2x + 1)/(2x)) + 3| + 1)^2 - 4(1|((2x + 1)/(2x)) + 3| + 1) + 2) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(3cos(1\u03c0(1|(1x + 5) + 1| + 2) - 3))^2 + 1(3cos(1\u03c0(1|(1x + 5) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-2(-1sin(1\u03c0x + 1) + 2)^2 + 1(-1sin(1\u03c0x + 1) + 2) + 1) + 2)/(1(-2(-1sin(1\u03c0x + 1) + 2)^2 + 1(-1sin(1\u03c0x + 1) + 2) + 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) = (2(-5(-2sin(1\u03c0x + 2) - 3) - 4) - 3)/(-1(-5(-2sin(1\u03c0x + 2) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(1|(-5(-2sin(3\u03c0x - 1) + 3) - 2)| - 3) - 1)/(-3(1|(-5(-2sin(3\u03c0x - 1) + 3) - 2)| - 3) - 3) 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|(4(-1cos(3\u03c0x - 2) - 3) + 2) - 3| - 1 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((-2(2cos(1\u03c0x + 1) - 1) + 1)/(-2(2cos(1\u03c0x + 1) - 1) - 3)) - 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) = (-1(-4(3sin(1\u03c0(-1|x + 1| + 2) - 1) + 3)^2 - 5(3sin(1\u03c0(-1|x + 1| + 2) - 1) + 3) - 4) + 1)/(1(-4(3sin(1\u03c0(-1|x + 1| + 2) - 1) + 3)^2 - 5(3sin(1\u03c0(-1|x + 1| + 2) - 1) + 3) - 4) - 2) 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) = 2sin(2\u03c0(-3|(-5x - 4) + 2| + 2) + 1) + 3 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) = 5(1|((2(1cos(2\u03c0x) + 3) + 2)/(3(1cos(2\u03c0x) + 3) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(2sin(2\u03c0(-2((-3x - 1)/(-3x + 1)) + 2) - 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) = 3sin(3\u03c0(4((3(-1|x + 2|))/(-1(-1|x + 2|) + 3)) + 4) - 1) + 1 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) = (2(2|(2cos(3\u03c0x + 3) - 1) - 1|) + 2)/(-2(2|(2cos(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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-1cos(2\u03c0((1x + 1)/(1x + 3)) + 3) + 3) + 3| - 2 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) = (1(-5(2|(-1sin(1\u03c0x - 2) - 1) + 1| - 1)^2 + 4(2|(-1sin(1\u03c0x - 2) - 1) + 1| - 1) + 1) + 3)/(-1(-5(2|(-1sin(1\u03c0x - 2) - 1) + 1| - 1)^2 + 4(2|(-1sin(1\u03c0x - 2) - 1) + 1| - 1) + 1) + 2) 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) = 4(-3cos(3\u03c0(-3x^2 - 5x)) - 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": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1|(2(1cos(3\u03c0x - 2) + 1)^2 + 2(1cos(3\u03c0x - 2) + 1) + 5) + 1|) + 2 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|(3(-3cos(2\u03c0x + 3) - 2)^2 + 5) - 3| + 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) = -1|(-2(-2sin(2\u03c0(-4x + 2)) - 3)^2 + 2(-2sin(2\u03c0(-4x + 2)) - 3) - 3) + 2| - 3 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) = 4(-2sin(1\u03c0(-1|x - 3| - 2)) - 2)^2 - 2 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) = 2cos(3\u03c0(2(-4(1|x + 2| - 1))^2 + 3(-4(1|x + 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": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-2cos(1\u03c0(-4(-5x + 2)^2 + 2(-5x + 2) - 2))) + 3)/(-1(-2cos(1\u03c0(-4(-5x + 2)^2 + 2(-5x + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2(-3cos(1\u03c0(1|x - 3| - 1) + 1) - 2) - 3) - 2)/(-1(2(-3cos(1\u03c0(1|x - 3| - 1) + 1) - 2) - 3) - 1) 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|(1(-3x - 5)^2 + 5(-3x - 5) - 5) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-2cos(3\u03c0(5x^2 - 1x - 4) - 3) + 1) + 2)/(-1(-2cos(3\u03c0(5x^2 - 1x - 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": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((1(-1sin(1\u03c0(-1|x - 2| + 1) - 2) + 1) + 1)/(2(-1sin(1\u03c0(-1|x - 2| + 1) - 2) + 1) - 3))^2 + 1((1(-1sin(1\u03c0(-1|x - 2| + 1) - 2) + 1) + 1)/(2(-1sin(1\u03c0(-1|x - 2| + 1) - 2) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((3(1sin(3\u03c0x - 1) - 2) + 1)/(1(1sin(3\u03c0x - 1) - 2) - 2)) - 2| - 1 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) = 2cos(3\u03c0(-2|(3(2x + 2)^2 - 2(2x + 2) - 4) - 2| + 1) - 1) - 1 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) = -3|(2cos(3\u03c0(3((2x - 2)/(3x - 2))^2 + 5((2x - 2)/(3x - 2)) + 5) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(-5(-2(1|x - 2| - 3) + 1)^2 - 4(-2(1|x - 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": "57", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-1|(4(-1sin(2\u03c0x + 2) - 2)^2 + 3(-1sin(2\u03c0x + 2) - 2) + 5) - 3| - 2) + 4 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(2|(-3(-3cos(3\u03c0x + 3) - 1)^2 - 5(-3cos(3\u03c0x + 3) - 1) + 2)| + 1) - 2)/(-1(2|(-3(-3cos(3\u03c0x + 3) - 1)^2 - 5(-3cos(3\u03c0x + 3) - 1) + 2)| + 1) + 3) negative?\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) = (-2(-2sin(1\u03c0(-3|(4x^2 - 2x + 3) + 1| + 2) - 3) + 1) - 1)/(2(-2sin(1\u03c0(-3|(4x^2 - 2x + 3) + 1| + 2) - 3) + 1) - 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(-1sin(1\u03c0(-2|x| - 3) + 2) + 1))/(2(-1sin(1\u03c0(-2|x| - 3) + 2) + 1) - 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) = -5(-3(3cos(3\u03c0((-3x + 3)/(2x + 3)) + 1) + 3) - 1)^2 - 5(-3(3cos(3\u03c0((-3x + 3)/(2x + 3)) + 1) + 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) = 1|(-2cos(1\u03c0((-2(-1x + 2) + 1)/(2(-1x + 2) + 3)) - 2) - 2) - 3| - 1 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) = -3(-3cos(2\u03c0(2|((1x + 2)/(2x)) + 3| - 2) - 2) + 2)^2 - 5(-3cos(2\u03c0(2|((1x + 2)/(2x)) + 3| - 2) - 2) + 2) - 2 is strictly decreasing.\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|(-1sin(3\u03c0((-1x - 2)/(1x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|((-2(-2cos(3\u03c0x - 1) - 1) - 2)/(-3(-2cos(3\u03c0x - 1) - 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": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((3(-3|(1cos(1\u03c0x) + 1)| + 2) + 1)/(2(-3|(1cos(1\u03c0x) + 1)| + 2) + 1))^2 + 4((3(-3|(1cos(1\u03c0x) + 1)| + 2) + 1)/(2(-3|(1cos(1\u03c0x) + 1)| + 2) + 1)) + 1 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(-2(-2sin(3\u03c0(-1|x + 2| - 1) - 3) - 3) + 5) + 2)/(-2(-2(-2sin(3\u03c0(-1|x + 2| - 1) - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1(-2cos(3\u03c0x - 1) + 3) + 4)^2 - 2(-1(-2cos(3\u03c0x - 1) + 3) + 4) + 5 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) = 1sin(1\u03c0(-4(-1x + 1)^2 + 4(-1x + 1) + 3) - 1) - 2 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) = -1(1sin(3\u03c0(3|x + 1|) - 3) + 2)^2 + 2(1sin(3\u03c0(3|x + 1|) - 3) + 2) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0(5(-2x - 3)^2 - 3(-2x - 3) - 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) = 1cos(1\u03c0(1|(-5(2x + 5)^2 + 5(2x + 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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-4(1sin(1\u03c0x - 2) - 3)^2 + 2(1sin(1\u03c0x - 2) - 3) - 2) + 1 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) = (3(2(2cos(1\u03c0x + 2))^2 + 4(2cos(1\u03c0x + 2)) - 5) + 3)/(1(2(2cos(1\u03c0x + 2))^2 + 4(2cos(1\u03c0x + 2)) - 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) = -1(-1|(2sin(3\u03c0x - 1) - 2) - 3|) + 5 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((-1(1cos(3\u03c0x) - 2) - 2)/(3(1cos(3\u03c0x) - 2) + 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) = -5(1|(1sin(1\u03c0x - 3) + 1) - 3| + 2) + 3 positive?\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(3cos(3\u03c0((3x - 3)/(-3x)) + 3)) + 3) + 1| + 1 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) = 2(-2cos(1\u03c0((1x + 1)/(1x - 2)) + 1) + 3)^2 + 5(-2cos(1\u03c0((1x + 1)/(1x - 2)) + 1) + 3) - 4 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) = -2(-1|(-2cos(1\u03c0x - 1) - 3) - 3| - 2)^2 - 5(-1|(-2cos(1\u03c0x - 1) - 3) - 3| - 2) + 2 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) = (3(-1sin(1\u03c0(5x^2 - 5x - 4)) - 1) + 2)/(2(-1sin(1\u03c0(5x^2 - 5x - 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": "35", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(1\u03c0(-2((2(1x + 1) - 1)/(-1(1x + 1)))^2 + 4((2(1x + 1) - 1)/(-1(1x + 1))) + 4) + 2) - 2 positive?\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) = (-1(-2sin(3\u03c0(-3|x + 2| - 1) - 2) + 2) + 1)/(1(-2sin(3\u03c0(-3|x + 2| - 1) - 2) + 2) + 2) negative?\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) = 5((-2(-3cos(1\u03c0x - 3) - 3) - 3)/(1(-3cos(1\u03c0x - 3) - 3) - 3)) - 4 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) = 4(-1sin(2\u03c0((2x + 3)/(-1x + 3)) + 3) - 1) + 5 positive?\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(3cos(2\u03c0(-1|x + 1| - 1) - 1) + 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) = -3sin(3\u03c0(-2|(-1(5x)^2 + 4(5x) - 2) + 1| - 1) - 2) + 1 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|(-2sin(3\u03c0((-1x + 2)/(1x + 1))) - 3) - 2| - 3 negative?\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) = -2sin(1\u03c0(-2|(-1(1x + 1)^2 + 5(1x + 1) + 5) - 1| - 2)) - 1 positive?\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(2|(-2cos(1\u03c0(5x^2 + 4x + 5)) + 1) - 3| - 1) + 5 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) = -4(1|(2sin(3\u03c0x + 3) - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1|(5(-2sin(3\u03c0x - 2) - 1) - 1)| - 2)^2 - 5(-1|(5(-2sin(3\u03c0x - 2) - 1) - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(2sin(3\u03c0((1x - 2)/(-1x - 2)) + 1)) + 2| is strictly increasing.\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|(2((-3(-2sin(1\u03c0x - 3) - 1) + 1)/(-1(-2sin(1\u03c0x - 3) - 1) + 3)) + 4)| + 2 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(-3|(-3(1cos(1\u03c0x + 3) - 2) + 3) + 3| + 3))/(1(-3|(-3(1cos(1\u03c0x + 3) - 2) + 3) + 3| + 3) - 3) 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) = -3sin(1\u03c0(1(-3|x - 1| - 2) + 3) - 3) - 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) = 1((-1(-3sin(2\u03c0(1|x - 1| + 1) - 3) + 2) - 3)/(-1(-3sin(2\u03c0(1|x - 1| + 1) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0(-1(-2|x - 2| + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(4(-2sin(1\u03c0(-1x^2 - 3x + 4) - 1) + 3) - 5) + 3)/(-2(4(-2sin(1\u03c0(-1x^2 - 3x + 4) - 1) + 3) - 5) + 1) 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) = (1(2cos(1\u03c0(4(-5x + 3)^2 + 4(-5x + 3) + 5)) + 2) + 2)/(1(2cos(1\u03c0(4(-5x + 3)^2 + 4(-5x + 3) + 5)) + 2) - 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) = 3|((-3(-1(3cos(1\u03c0x - 2) - 1)^2 + 1(3cos(1\u03c0x - 2) - 1) + 4) - 2)/(2(-1(3cos(1\u03c0x - 2) - 1)^2 + 1(3cos(1\u03c0x - 2) - 1) + 4) - 1)) - 3| + 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) = (-2(1cos(3\u03c0(3(1|x + 2| + 3)^2 - 5(1|x + 2| + 3) + 3)) + 1) - 3)/(1(1cos(3\u03c0(3(1|x + 2| + 3)^2 - 5(1|x + 2| + 3) + 3)) + 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) = -4((1(1(-2sin(2\u03c0x) - 2)^2 - 4(-2sin(2\u03c0x) - 2) - 3) - 3)/(-1(1(-2sin(2\u03c0x) - 2)^2 - 4(-2sin(2\u03c0x) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-5(-1(1sin(1\u03c0x - 3))^2 - 2(1sin(1\u03c0x - 3))) - 5) - 3| + 2 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) = (1(3cos(2\u03c0(2x + 1) + 3)) - 3)/(1(3cos(2\u03c0(2x + 1) + 3)) - 2) 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) = -4(3|(-1cos(2\u03c0(-2x^2 + 2x - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(-1|(1x + 2) + 3| + 1) - 1) + 1 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) = 2sin(3\u03c0(1(-1((-1x + 2)/(-1x - 3))^2 + 4((-1x + 2)/(-1x - 3)) - 5) - 5)) + 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) = 4(-1|(1(2cos(3\u03c0x + 1) + 2) + 4) + 2| + 1)^2 + 4(-1|(1(2cos(3\u03c0x + 1) + 2) + 4) + 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) = -1cos(1\u03c0(3(-5x - 2)^2 + 3(-5x - 2) + 3) - 2) - 2 is strictly increasing.\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(-1cos(3\u03c0x + 1) - 1)^2 - 2(-1cos(3\u03c0x + 1) - 1) - 5) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3cos(2\u03c0(-5x - 1) - 3) - 1)^2 + 1(-3cos(2\u03c0(-5x - 1) - 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": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(2(1sin(3\u03c0(3x^2 + 5x + 5) - 1) + 1) + 5) + 3| - 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) = 1sin(3\u03c0(-1|(-5x) + 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": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(4(-2sin(3\u03c0x + 1) - 1)) - 3)/(3(4(-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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3cos(2\u03c0(-3x + 5) - 3) - 1) + 2| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-3(-1sin(2\u03c0(2|x + 1| - 1) - 1) - 3) - 4)^2 - 5(-3(-1sin(2\u03c0(2|x + 1| - 1) - 1) - 3) - 4) + 1 is strictly increasing.\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|(1sin(3\u03c0(2x^2 - 1x - 3) + 1) + 1) + 2| 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((1(-1|(3cos(3\u03c0x + 1) + 2) + 3| - 1) + 2)/(-3(-1|(3cos(3\u03c0x + 1) + 2) + 3| - 1) - 1))^2 - 3((1(-1|(3cos(3\u03c0x + 1) + 2) + 3| - 1) + 2)/(-3(-1|(3cos(3\u03c0x + 1) + 2) + 3| - 1) - 1)) + 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) = 3cos(3\u03c0(5(-2|x - 3| + 2)^2 + 4(-2|x - 3| + 2) - 2) - 1) + 3 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) = 1(1(-1sin(2\u03c0x + 2) + 1)^2 + 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) = -4(3sin(2\u03c0(-3((-2x + 1)/(-2x - 3))^2 - 3((-2x + 1)/(-2x - 3)) - 2) + 1) - 3) + 5 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) = 4(-1|(-3(-2cos(3\u03c0x - 2) - 2) - 5) - 1| - 3)^2 - 3(-1|(-3(-2cos(3\u03c0x - 2) - 2) - 5) - 1| - 3) - 5 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) = -1sin(2\u03c0((-2(1|(1x^2 - 5x + 4) - 2| - 1) + 3)/(1(1|(1x^2 - 5x + 4) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-3(1sin(2\u03c0(-2|x| - 1) - 3) - 1) + 1))/(2(-3(1sin(2\u03c0(-2|x| - 1) - 3) - 1) + 1) + 3) 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) = (1(3cos(3\u03c0(-3|(-1x^2 - 2x - 3) - 2| - 3) - 1) - 3) + 1)/(-2(3cos(3\u03c0(-3|(-1x^2 - 2x - 3) - 2| - 3) - 1) - 3) + 3) 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(1(1sin(1\u03c0x) - 3) + 4)^2 - 1(1(1sin(1\u03c0x) - 3) + 4) + 3 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(3\u03c0(-5(1|((-2x)/(-3x + 1)) - 1|) + 3) - 1) - 3 negative?\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) = -2|((1(1sin(1\u03c0x + 1) + 1))/(-1(1sin(1\u03c0x + 1) + 1) - 2)) + 3| + 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) = 1|(2(2cos(1\u03c0x - 2) + 3) - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(2|(-1cos(2\u03c0(2x^2 + 5) - 2) - 3) + 1|) + 1)/(2(2|(-1cos(2\u03c0(2x^2 + 5) - 2) - 3) + 1|) + 3) is strictly decreasing.\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(-4((-1(-1x + 4) - 3)/(-2(-1x + 4) + 3))^2 - 5((-1(-1x + 4) - 3)/(-2(-1x + 4) + 3)) - 3) + 1) - 3 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) = (-3(3|(3(-1sin(3\u03c0x - 1) - 2) + 4) - 1| + 2) + 1)/(2(3|(3(-1sin(3\u03c0x - 1) - 2) + 4) - 1| + 2) - 3) 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) = -5(3cos(2\u03c0(3|x + 2| - 1) - 1) - 3)^2 - 1(3cos(2\u03c0(3|x + 2| - 1) - 1) - 3) - 4 is strictly increasing.\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(-1(2cos(2\u03c0x + 1) + 2) - 4)^2 - 1(-1(2cos(2\u03c0x + 1) + 2) - 4) - 2) - 1)/(1(-1(-1(2cos(2\u03c0x + 1) + 2) - 4)^2 - 1(-1(2cos(2\u03c0x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2|(2sin(3\u03c0x - 1) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-5(2(3sin(3\u03c0x + 2) + 2)^2 + 5(3sin(3\u03c0x + 2) + 2) - 1) + 3) + 2)/(2(-5(2(3sin(3\u03c0x + 2) + 2)^2 + 5(3sin(3\u03c0x + 2) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1(2sin(2\u03c0x - 3) - 3)^2 + 1(2sin(2\u03c0x - 3) - 3) - 5) - 3| - 1 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) = -1sin(1\u03c0(2(2(-3|x - 1|)^2 + 1(-3|x - 1|)) - 1) - 2) + 3 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(-2cos(2\u03c0(-4(-1x - 2)^2 - 5(-1x - 2)) + 2) + 1) + 3)/(2(-2cos(2\u03c0(-4(-1x - 2)^2 - 5(-1x - 2)) + 2) + 1) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-1sin(1\u03c0(-4x^2 - 4x + 4) + 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": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(1cos(2\u03c0(4x + 2) + 3) - 1))/(2(1cos(2\u03c0(4x + 2) + 3) - 1) - 1) 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) = (-2(1(-3sin(3\u03c0x) + 1)^2 + 5(-3sin(3\u03c0x) + 1) - 4))/(-3(1(-3sin(3\u03c0x) + 1)^2 + 5(-3sin(3\u03c0x) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2(3sin(3\u03c0(-3x^2 - 4x - 4) + 1) + 2) - 4) - 3| 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) = (-1(2sin(3\u03c0(5x^2 - 4)) + 1) - 2)/(-2(2sin(3\u03c0(5x^2 - 4)) + 1) - 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) = -4(1sin(3\u03c0(1|x + 1| + 1) - 3))^2 + 4(1sin(3\u03c0(1|x + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(4(-1x^2 + 2x + 1) + 3) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(-3(1(3|x + 2| - 3)^2 - 3(3|x + 2| - 3) + 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": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((1(3(1sin(2\u03c0x + 2) - 3) - 2) - 1)/(-2(3(1sin(2\u03c0x + 2) - 3) - 2) + 3)) - 1| + 2 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) = 1|(-2(3cos(3\u03c0x + 2) - 3)^2 - 4(3cos(3\u03c0x + 2) - 3) + 1) + 1| + 2 negative?\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) = 2(-3|(2sin(2\u03c0(-3x))) + 3| - 3)^2 + 2(-3|(2sin(2\u03c0(-3x))) + 3| - 3) - 5 negative?\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(3\u03c0(-1((1x - 1)/(1x + 1))^2 + 5((1x - 1)/(1x + 1)) - 4) - 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": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1|(-3sin(2\u03c0(-1x^2 + 5x + 1) + 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": "55", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1sin(1\u03c0(2|x + 3| + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(3(4x^2 + 5x - 5) - 5) + 3) - 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) = (2(4(1(-2sin(2\u03c0x + 2) + 3)^2 - 1(-2sin(2\u03c0x + 2) + 3) + 1) - 4) + 1)/(1(4(1(-2sin(2\u03c0x + 2) + 3)^2 - 1(-2sin(2\u03c0x + 2) + 3) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(4(1sin(1\u03c0x - 3) - 1)^2 - 1(1sin(1\u03c0x - 3) - 1) + 1) + 3)/(-1(4(1sin(1\u03c0x - 3) - 1)^2 - 1(1sin(1\u03c0x - 3) - 1) + 1) + 2) negative?\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|(-2(-1cos(2\u03c0x - 2) - 2)) - 3| + 3 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) = 2((-2(-1|(-3cos(2\u03c0x + 1) + 1) - 2| + 3) + 2)/(3(-1|(-3cos(2\u03c0x + 1) + 1) - 2| + 3) + 2))^2 - 3((-2(-1|(-3cos(2\u03c0x + 1) + 1) - 2| + 3) + 2)/(3(-1|(-3cos(2\u03c0x + 1) + 1) - 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": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-3sin(3\u03c0(-2(-3|x + 3| - 2) - 3) + 2) - 2)^2 - 2(-3sin(3\u03c0(-2(-3|x + 3| - 2) - 3) + 2) - 2) + 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(-1cos(1\u03c0(3x^2 - 4x + 3) - 1) - 1) + 1)/(2(-1cos(1\u03c0(3x^2 - 4x + 3) - 1) - 1) + 1) positive?\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) = -3|(2cos(2\u03c0((2x + 3)/(-1x + 3))) + 2) + 1| + 2 is strictly increasing.\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) = 2((1(1|(-2sin(3\u03c0x + 1) + 1) + 3| + 3) - 2)/(2(1|(-2sin(3\u03c0x + 1) + 1) + 3| + 3) - 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) = -2cos(1\u03c0(3|(3x + 1) - 2| - 3) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(1|(5((3x + 1)/(-1x - 2))^2 + 5((3x + 1)/(-1x - 2)) - 4)| - 2) + 2) + 3 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(-1|(-1cos(2\u03c0x - 2) - 1) - 2|)^2 + 4(-1|(-1cos(2\u03c0x - 2) - 1) - 2|) - 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) = 4(-1|(-2sin(1\u03c0x - 3) - 3) + 2| - 1)^2 + 4(-1|(-2sin(1\u03c0x - 3) - 3) + 2| - 1) - 5 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|(-3sin(2\u03c0(-3x^2 - 3x + 5) + 3) - 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) = (-1(-1sin(1\u03c0(2|x + 1| + 1) + 3) + 1) + 2)/(-1(-1sin(1\u03c0(2|x + 1| + 1) + 3) + 1) - 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) = (3(2cos(1\u03c0(2x - 2) - 1) + 2))/(2(2cos(1\u03c0(2x - 2) - 1) + 2) + 3) 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) = -1cos(2\u03c0(2((-2(2|x| - 3) - 1)/(-2(2|x| - 3) + 1))^2 + 2((-2(2|x| - 3) - 1)/(-2(2|x| - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-5(1sin(1\u03c0x + 3) - 3)^2 - 4(1sin(1\u03c0x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0(-1((-1x + 2)/(1x + 1))^2 + 2((-1x + 2)/(1x + 1)) + 5)) - 2 is strictly decreasing.\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) = (-1(-4(1sin(3\u03c0(1x^2 - 4x - 1) + 2))) - 2)/(2(-4(1sin(3\u03c0(1x^2 - 4x - 1) + 2)))) positive?\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) = 2|(3(2sin(2\u03c0x + 1) + 2) - 1)| + 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) = 1|(1(-1cos(2\u03c0x - 2) - 2) - 1) - 1| + 3 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) = -1(-3(-1sin(3\u03c0x + 1) + 1) - 1)^2 + 5(-3(-1sin(3\u03c0x + 1) + 1) - 1) + 5 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) = 2|((-1(-1sin(3\u03c0x + 3) + 3))/(3(-1sin(3\u03c0x + 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": "30", "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 - 4) - 3| + 3) - 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((-1(1cos(2\u03c0x + 2) - 3) + 1)/(-3(1cos(2\u03c0x + 2) - 3) - 3)) + 4) + 1| - 3 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) = 2cos(1\u03c0(1|((-3x - 1)/(-1x - 3)) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0(-5(2(2|x - 2|) + 1)^2 + 3(2(2|x - 2|) + 1) + 5)) + 1 negative?\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) = 2sin(2\u03c0(-3|((-3(-3x - 1))/(-1(-3x - 1) - 2)) - 3| + 2) - 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(2|(2sin(2\u03c0(1x - 2) - 3) + 1) + 1| - 2) + 1)/(-3(2|(2sin(2\u03c0(1x - 2) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(1sin(1\u03c0(3x^2 + 3x - 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": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-5(3cos(1\u03c0x) + 3)^2 + 4(3cos(1\u03c0x) + 3) + 5) + 3)/(-2(-5(3cos(1\u03c0x) + 3)^2 + 4(3cos(1\u03c0x) + 3) + 5) + 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) = 4(-3(3cos(1\u03c0x + 1) - 2) - 2)^2 - 1(-3(3cos(1\u03c0x + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(2\u03c0(1((-1x + 3)/(1x + 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": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1sin(3\u03c0((3x + 3)/(-2x + 3)) + 1) + 2) - 3| - 2 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) = (-1(3cos(3\u03c0(-2|x| - 3) + 3) - 1) + 2)/(1(3cos(3\u03c0(-2|x| - 3) + 3) - 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) = 1(-2|(2sin(2\u03c0x - 1) - 2) + 1| + 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) = 1sin(3\u03c0(-4(1|x - 3| - 1)^2 + 5(1|x - 3| - 1) + 1)) + 1 negative?\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) = -1(2cos(1\u03c0(-3|x - 3| + 2)) + 3) + 1 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(-3|(-2cos(2\u03c0x - 3) + 3)|) + 2)/(1(-3|(-2cos(2\u03c0x - 3) + 3)|) - 3) 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) = -4(-3sin(3\u03c0((2(4x^2 + 5x + 2) - 3)/(1(4x^2 + 5x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2cos(2\u03c0(-4x^2 + 1x - 3) + 1) + 3))/(3(2cos(2\u03c0(-4x^2 + 1x - 3) + 1) + 3) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0((-2(-1x + 3) + 3)/(2(-1x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(4(1|((-1x + 3)/(1x)) - 2| - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(1(-2cos(3\u03c0(1x^2 - 5x - 5) - 2) - 2) - 5) - 3)/(3(1(-2cos(3\u03c0(1x^2 - 5x - 5) - 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": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2(2sin(2\u03c0x - 1) + 3)^2 - 5(2sin(2\u03c0x - 1) + 3) + 3) + 4 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) = -2(-1cos(3\u03c0(-1|((2x + 2)/(-1x - 2)) - 2|) - 1) - 3)^2 + 5(-1cos(3\u03c0(-1|((2x + 2)/(-1x - 2)) - 2|) - 1) - 3) + 5 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(3sin(2\u03c0(-3|x + 2| - 3) - 3)) + 2)/(3(3sin(2\u03c0(-3|x + 2| - 3) - 3))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((-1(-1cos(2\u03c0(4x^2 - 5x) - 3) - 1) + 1)/(2(-1cos(2\u03c0(4x^2 - 5x) - 3) - 1) + 1)) 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) = (-1(-3|(2cos(2\u03c0x + 1) + 1) + 1| + 3))/(1(-3|(2cos(2\u03c0x + 1) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-1|(-2(1sin(2\u03c0x - 1) + 1)^2 + 5(1sin(2\u03c0x - 1) + 1) - 2) + 2| + 1) + 5 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) = 1sin(3\u03c0(4(2|(-1x - 4) - 1|)^2 - 2(2|(-1x - 4) - 1|) - 1) + 1) - 1 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) = -5(-2cos(1\u03c0((-1(1|x - 2| - 2) - 3)/(3(1|x - 2| - 2) + 3)) - 3) + 2)^2 + 5(-2cos(1\u03c0((-1(1|x - 2| - 2) - 3)/(3(1|x - 2| - 2) + 3)) - 3) + 2) + 2 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) = (3(2(5(3cos(1\u03c0x - 2) + 1))^2 + 4(5(3cos(1\u03c0x - 2) + 1)) - 3))/(-2(2(5(3cos(1\u03c0x - 2) + 1))^2 + 4(5(3cos(1\u03c0x - 2) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-4(-3sin(2\u03c0x + 2) + 2)^2 + 4(-3sin(2\u03c0x + 2) + 2)) + 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) = -1|((1(2cos(1\u03c0x - 2) - 2) + 2)/(1(2cos(1\u03c0x - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(5((-3x - 1)/(1x - 2))^2 + 2((-3x - 1)/(1x - 2)) - 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": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(1sin(1\u03c0(1(1|x + 2| - 1)^2 + 3(1|x + 2| - 1) + 1)) - 2) - 3)/(1(1sin(1\u03c0(1(1|x + 2| - 1)^2 + 3(1|x + 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(3cos(3\u03c0(-2|((1x - 1)/(2x))| + 3) - 2) + 1) + 1 is strictly increasing.\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(3cos(2\u03c0(1x - 3) - 2) - 1)^2 - 5(3cos(2\u03c0(1x - 3) - 2) - 1) - 4 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(1sin(3\u03c0(4x^2 + 3x - 2))) + 1 is strictly increasing.\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) = -3cos(2\u03c0(-3|(2x^2) - 2| + 3) - 1) - 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) = -5(3cos(1\u03c0(-1x - 4)) - 3)^2 + 2(3cos(1\u03c0(-1x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-2(-1cos(3\u03c0x - 1) - 3)^2 - 5(-1cos(3\u03c0x - 1) - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3\u03c0(5(-1x^2 + 4x + 2)) - 1) + 1 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) = -5(-1|(-3sin(3\u03c0x + 3)) + 1| - 2)^2 - 3(-1|(-3sin(3\u03c0x + 3)) + 1| - 2) - 4 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) = 3(2|(-4(-2sin(1\u03c0x + 2) - 2) - 4) + 3| + 1)^2 - 1(2|(-4(-2sin(1\u03c0x + 2) - 2) - 4) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3sin(2\u03c0((-1(-1|x| + 1) - 1)/(2(-1|x| + 1) + 1)) - 2) - 2) - 5 is strictly increasing.\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) = 5(3|((-1(-2cos(2\u03c0x + 1) - 1) + 3)/(-1(-2cos(2\u03c0x + 1) - 1) + 2)) - 2| + 2) - 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) = -2(2sin(1\u03c0(2|x| + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-3sin(2\u03c0(-5(-1|x + 1|)) - 3) + 2) - 2)/(-1(-3sin(2\u03c0(-5(-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": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3(-1sin(2\u03c0x + 3)) - 5) - 3| + 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) = -1cos(3\u03c0(2|((-3(-3x^2 + 4))/(1(-3x^2 + 4) + 1)) + 3| - 3) + 1) - 2 is strictly increasing.\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) = -1cos(1\u03c0((-1(2(1|x + 3|)^2 - 1(1|x + 3|) - 4) - 3)/(1(2(1|x + 3|)^2 - 1(1|x + 3|) - 4) + 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(-2|(2(3sin(3\u03c0x - 1) + 2)^2 + 1(3sin(3\u03c0x - 1) + 2) + 5) + 3| - 3) + 2)/(-1(-2|(2(3sin(3\u03c0x - 1) + 2)^2 + 1(3sin(3\u03c0x - 1) + 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": "45", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(1|(4x + 3)| + 3) - 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0((-1(1x^2 + 2x - 2) - 2)/(-1(1x^2 + 2x - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3\u03c0(1|(-1x - 1) + 1| - 1) - 1) - 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) = -1(-3sin(1\u03c0(5x - 4) - 1) + 2)^2 + 3(-3sin(1\u03c0(5x - 4) - 1) + 2) + 2 positive?\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) = 1sin(2\u03c0(1(1|x| - 2))) - 3 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) = -1(-1|(-1(-1sin(3\u03c0x + 1) - 1) + 3) + 3| - 1)^2 - 2(-1|(-1(-1sin(3\u03c0x + 1) - 1) + 3) + 3| - 1) - 2 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) = 3(2sin(3\u03c0(5(-1|x - 2| - 1) + 3) + 3) - 3)^2 - 2(2sin(3\u03c0(5(-1|x - 2| - 1) + 3) + 3) - 3) - 4 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) = 1cos(3\u03c0(3(1|((2x + 3)/(-2x + 3))| + 3)^2 - 4(1|((2x + 3)/(-2x + 3))| + 3) + 5) + 1) + 3 negative?\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) = 4(1(-1sin(2\u03c0x + 2) - 3) - 2)^2 - 4(1(-1sin(2\u03c0x + 2) - 3) - 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) = 2|(4(2cos(2\u03c0(-5x - 4) - 3) + 2)^2 - 1(2cos(2\u03c0(-5x - 4) - 3) + 2) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-1cos(3\u03c0(-2(2x + 2)^2 - 1(2x + 2) - 4) - 1) - 3) + 1)/(1(-1cos(3\u03c0(-2(2x + 2)^2 - 1(2x + 2) - 4) - 1) - 3) - 3) 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) = 2cos(3\u03c0(-1|(-1x + 5) - 1| + 1) + 3) - 3 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) = -3cos(1\u03c0(-2(-2|(2x^2 + 4) - 2| - 1) - 3) - 1) - 3 negative?\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(5(2cos(1\u03c0x - 1) + 2)^2 - 1(2cos(1\u03c0x - 1) + 2) - 3) - 3)/(3(5(2cos(1\u03c0x - 1) + 2)^2 - 1(2cos(1\u03c0x - 1) + 2) - 3) + 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(-1|(-1cos(2\u03c0(1x + 2) - 2) + 1)| + 3)^2 - 1(-1|(-1cos(2\u03c0(1x + 2) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2|((-3(1sin(2\u03c0x - 3) - 1) - 1)/(1(1sin(2\u03c0x - 3) - 1) + 1))| - 3)^2 - 2(-2|((-3(1sin(2\u03c0x - 3) - 1) - 1)/(1(1sin(2\u03c0x - 3) - 1) + 1))| - 3) - 1 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) = -1(3cos(3\u03c0(2|((1x + 3)/(1x + 2)) - 1|) + 3) + 3)^2 + 5(3cos(3\u03c0(2|((1x + 3)/(1x + 2)) - 1|) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((3(1cos(3\u03c0(4x - 3) - 1)) + 2)/(2(1cos(3\u03c0(4x - 3) - 1)) - 1))^2 + 5((3(1cos(3\u03c0(4x - 3) - 1)) + 2)/(2(1cos(3\u03c0(4x - 3) - 1)) - 1)) + 2 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) = 2|(1cos(2\u03c0((-1x + 1)/(-1x + 2)) + 3)) - 3| + 1 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) = -1sin(1\u03c0(2(-2|(3x - 1) - 3| - 3)^2 - 5(-2|(3x - 1) - 3| - 3) + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0(-3|(2x - 5) + 3| - 1)) + 2 positive?\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(3sin(3\u03c0(-1|(-3x + 4) + 3| + 1)) + 3)^2 - 1(3sin(3\u03c0(-1|(-3x + 4) + 3| + 1)) + 3) + 1 is strictly increasing.\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) = 3|(2sin(3\u03c0((3(-3x^2 - 3x - 2) + 2)/(-1(-3x^2 - 3x - 2) - 1)) + 1)) - 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) = -3(2|(3sin(1\u03c0x) - 3) - 2| + 1)^2 - 5 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) = (-3(1cos(3\u03c0(-1|x - 3| - 2) + 2) - 2))/(2(1cos(3\u03c0(-1|x - 3| - 2) + 2) - 2) + 1) 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((-3(-2sin(2\u03c0x - 2) + 3) + 1)/(-1(-2sin(2\u03c0x - 2) + 3) + 2)) - 3 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) = 4(1|((-1(1sin(1\u03c0x - 2)))/(-1(1sin(1\u03c0x - 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": "For the function f(x) = 1(-1|(-3cos(2\u03c0x + 2) + 2) - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-4(1sin(1\u03c0x - 2) - 1) - 4) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(1(-1sin(1\u03c0x + 3) - 1)^2 - 1(-1sin(1\u03c0x + 3) - 1) + 5)| + 1 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) = 1|(-2sin(3\u03c0(1x + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3(-1|(-3cos(2\u03c0x - 1) - 1) + 1| - 2) + 1)^2 + 3(-3(-1|(-3cos(2\u03c0x - 1) - 1) + 1| - 2) + 1) + 5 negative?\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|(-1sin(1\u03c0(4x + 1)) + 1) + 3| - 2 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) = 3|(-4(2sin(1\u03c0x - 2) - 1) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2(-1(2cos(1\u03c0x + 3) + 1)^2 - 5(2cos(1\u03c0x + 3) + 1) + 1) - 1) + 3)/(2(2(-1(2cos(1\u03c0x + 3) + 1)^2 - 5(2cos(1\u03c0x + 3) + 1) + 1) - 1) + 2) is strictly decreasing.\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((-1(-2|(-1cos(1\u03c0x) - 1) - 2| + 1) + 1)/(3(-2|(-1cos(1\u03c0x) - 1) - 2| + 1) + 2))^2 + 2((-1(-2|(-1cos(1\u03c0x) - 1) - 2| + 1) + 1)/(3(-2|(-1cos(1\u03c0x) - 1) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3|(-2cos(3\u03c0((-3x + 2)/(1x + 1)) + 1)) + 2| + 2)^2 + 2(3|(-2cos(3\u03c0((-3x + 2)/(1x + 1)) + 1)) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-1(3cos(1\u03c0(3x + 1) - 1) - 3) - 3)/(-1(3cos(1\u03c0(3x + 1) - 1) - 3) - 1))^2 - 5((-1(3cos(1\u03c0(3x + 1) - 1) - 3) - 3)/(-1(3cos(1\u03c0(3x + 1) - 1) - 3) - 1)) + 1 negative?\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) = (-2(-1|(-2cos(3\u03c0x + 1)) + 1| + 1) + 2)/(1(-1|(-2cos(3\u03c0x + 1)) + 1| + 1)) negative?\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) = -1(-3|(-2sin(3\u03c0x - 1) + 2) - 2| - 3)^2 - 2(-3|(-2sin(3\u03c0x - 1) + 2) - 2| - 3) + 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(1sin(3\u03c0((2x - 1)/(-1x)) - 3) + 3)^2 - 5(1sin(3\u03c0((2x - 1)/(-1x)) - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-4(4(-2sin(2\u03c0x + 1) + 1) + 5)^2 + 5)| + 3 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(2\u03c0(1((-1x + 1)/(-1x))^2 + 1((-1x + 1)/(-1x)) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-3cos(1\u03c0(-1|x + 3| + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3sin(2\u03c0((3x)/(1x - 2)) + 3))^2 - 5(-3sin(2\u03c0((3x)/(1x - 2)) + 3)) + 3 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(1sin(3\u03c0(2|(5x - 1) - 3| + 3) + 1) - 2) - 3)/(-1(1sin(3\u03c0(2|(5x - 1) - 3| + 3) + 1) - 2) + 2) is strictly decreasing.\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(1x^2 - 4x - 3)) + 2) + 3 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(3(3cos(1\u03c0x - 2) + 2) + 5)^2 + 5(3(3cos(1\u03c0x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(3((1x + 1)/(2x - 1))^2 - 3((1x + 1)/(2x - 1)) + 5) - 3) - 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) = 3|(1cos(1\u03c0(-3x - 1) + 1) - 2) - 1| + 3 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) = 2|(4(-3sin(2\u03c0x + 3)) - 1) - 3| + 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) = (-1(-3(-1sin(3\u03c0(-2x^2 + 4x) + 2) + 2) + 3) + 1)/(-1(-3(-1sin(3\u03c0(-2x^2 + 4x) + 2) + 2) + 3) + 2) 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) = -1sin(3\u03c0(-4(2|x + 1| + 1)^2 + 2(2|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": "26", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(1|(-3sin(3\u03c0((-1x - 1)/(-3x + 1)) - 3) + 3) - 1|)^2 - 4(1|(-3sin(3\u03c0((-1x - 1)/(-3x + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-2cos(1\u03c0(1x^2 - 2x + 3) - 1) - 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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-4(-2cos(1\u03c0((3x + 3)/(1x - 2)) + 2) + 2)^2 - 2(-2cos(1\u03c0((3x + 3)/(1x - 2)) + 2) + 2) - 5) - 3 is strictly decreasing.\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(-1sin(1\u03c0(4(-3|x - 1|) + 4) + 2) + 2)^2 + 3(-1sin(1\u03c0(4(-3|x - 1|) + 4) + 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": "77", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-3cos(3\u03c0(-5(-3x^2 - 1x + 1) + 3) + 3) + 1) + 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) = -4(1(-3|(-2cos(1\u03c0x) - 3) + 2| - 2) + 1)^2 + 1(1(-3|(-2cos(1\u03c0x) - 3) + 2| - 2) + 1) + 4 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) = -3|(-3cos(2\u03c0(4(-2x - 5)^2 - 2(-2x - 5) + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-3|(3cos(3\u03c0(5x^2 + 3x + 3) - 1) + 3)| - 3) + 1 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(4(-1(2cos(3\u03c0x + 3) + 1) - 4)^2 + 3(-1(2cos(3\u03c0x + 3) + 1) - 4) + 1) + 2)/(-2(4(-1(2cos(3\u03c0x + 3) + 1) - 4)^2 + 3(-1(2cos(3\u03c0x + 3) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-4(3cos(3\u03c0x - 2) + 2) - 1)^2 + 4(-4(3cos(3\u03c0x - 2) + 2) - 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-2(-2(1sin(2\u03c0x - 1) - 3))^2 - 2(-2(1sin(2\u03c0x - 1) - 3)) - 3) + 3)/(3(-2(-2(1sin(2\u03c0x - 1) - 3))^2 - 2(-2(1sin(2\u03c0x - 1) - 3)) - 3) + 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) = -5(3|((3(-2sin(3\u03c0x) - 3) - 2)/(-3(-2sin(3\u03c0x) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3\u03c0(-3(3(-1|x + 1| + 2) + 4)^2 + 2(3(-1|x + 1| + 2) + 4) + 2) + 2) + 2 is strictly increasing.\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) = 1cos(3\u03c0(2|((-1x - 2)/(2x - 3)) + 2| + 1) - 2) + 1 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) = (-3(-3|(-2sin(3\u03c0(3x - 4) - 3) - 3) - 3| - 1) - 1)/(-3(-3|(-2sin(3\u03c0(3x - 4) - 3) - 3) - 3| - 1) + 4) is strictly decreasing.\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) = -3|(-2(2cos(2\u03c0x) - 3) - 3) + 1| - 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) = (1(-2(-3sin(1\u03c0x - 3) + 3)^2 - 4(-3sin(1\u03c0x - 3) + 3) - 2) + 2)/(-1(-2(-3sin(1\u03c0x - 3) + 3)^2 - 4(-3sin(1\u03c0x - 3) + 3) - 2)) 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) = -4((-3(1sin(1\u03c0(5x - 2) - 1) + 2) - 2)/(3(1sin(1\u03c0(5x - 2) - 1) + 2) - 2))^2 - 1((-3(1sin(1\u03c0(5x - 2) - 1) + 2) - 2)/(3(1sin(1\u03c0(5x - 2) - 1) + 2) - 2)) - 4 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(-3cos(1\u03c0x + 3)) + 1)^2 + 5(5(-3cos(1\u03c0x + 3)) + 1) - 4 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(2sin(3\u03c0(3(-1x)^2 - 2(-1x) + 4) + 2) - 3) + 3)/(2(2sin(3\u03c0(3(-1x)^2 - 2(-1x) + 4) + 2) - 3) + 3) is strictly decreasing.\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) = 3|(2sin(2\u03c0(-4((1x)/(-2x + 3))) - 1)) - 3| + 1 is strictly increasing.\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) = -2cos(1\u03c0((2(1|(5x^2 + 5x - 2)| - 2) - 1)/(1(1|(5x^2 + 5x - 2)| - 2) - 2)) - 3) - 2 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) = 2cos(2\u03c0(4(2|x - 2|)^2 + 2(2|x - 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": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3((3(-1cos(3\u03c0x - 1) + 2) - 1)/(-1(-1cos(3\u03c0x - 1) + 2) - 1))^2 + 3((3(-1cos(3\u03c0x - 1) + 2) - 1)/(-1(-1cos(3\u03c0x - 1) + 2) - 1)) + 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) = -1|(4(-2sin(1\u03c0(-2x^2 + 2x - 2)) - 2) - 3) + 2| - 1 is strictly increasing.\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(-5((3x + 1)/(2x + 2)) - 4) - 2) - 2)^2 + 1(-2cos(1\u03c0(-5((3x + 1)/(2x + 2)) - 4) - 2) - 2) + 1 is strictly decreasing.\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|(1sin(2\u03c0(5(5x)^2 - 5(5x) - 5) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-4((1(2sin(1\u03c0x + 2)) - 1)/(1(2sin(1\u03c0x + 2)))) + 4)^2 - 1(-4((1(2sin(1\u03c0x + 2)) - 1)/(1(2sin(1\u03c0x + 2)))) + 4) + 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) = -2cos(3\u03c0(-3(5((-2x)/(2x + 2)) + 5)^2 + 3(5((-2x)/(2x + 2)) + 5) + 4) - 1) positive?\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) = 3sin(2\u03c0((-1(-3x^2 + 4x + 5) + 1)/(2(-3x^2 + 4x + 5) + 1)) + 1) + 3 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) = (1(2(1sin(3\u03c0x + 2) - 2) - 3))/(2(2(1sin(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((2(3cos(1\u03c0x + 1) - 1) - 1)/(3(3cos(1\u03c0x + 1) - 1) + 3))^2 - 2((2(3cos(1\u03c0x + 1) - 1) - 1)/(3(3cos(1\u03c0x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-1(-3|(2sin(1\u03c0x + 1) + 1) + 3| + 1)^2 + 2(-3|(2sin(1\u03c0x + 1) + 1) + 3| + 1) + 5) - 1 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) = 1sin(1\u03c0(1((3x - 3)/(-2x))^2 + 1((3x - 3)/(-2x)) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(-2(-3((-2x)/(1x + 3)) - 4)^2 - 3(-3((-2x)/(1x + 3)) - 4) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(2\u03c0(-4((-3(1|x + 1| - 3) + 3)/(-3(1|x + 1| - 3) + 4))^2 + 2((-3(1|x + 1| - 3) + 3)/(-3(1|x + 1| - 3) + 4)) - 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": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-3cos(1\u03c0(-2x + 2) + 3) + 2) + 3)/(3(-3cos(1\u03c0(-2x + 2) + 3) + 2) - 2) 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|(-1(2sin(3\u03c0(-1x + 5) + 1) - 1)^2 - 3(2sin(3\u03c0(-1x + 5) + 1) - 1) + 4) - 2| + 2 negative?\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) = (3(3(-2cos(1\u03c0x - 2) + 2)^2 - 4(-2cos(1\u03c0x - 2) + 2) - 2) - 3)/(-3(3(-2cos(1\u03c0x - 2) + 2)^2 - 4(-2cos(1\u03c0x - 2) + 2) - 2) + 2) 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) = (-1(2(-2cos(2\u03c0x + 2) + 2)^2 - 5) - 1)/(-3(2(-2cos(2\u03c0x + 2) + 2)^2 - 5) + 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) = 3cos(2\u03c0(-1|(4((-2x + 2)/(1x))^2 - 4((-2x + 2)/(1x)) + 4) - 2| - 3) + 1) + 2 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) = (-1(-2cos(1\u03c0(4x^2 - 2x - 1)) + 1) - 2)/(1(-2cos(1\u03c0(4x^2 - 2x - 1)) + 1) - 3) negative?\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) = -1sin(2\u03c0(-2(5x)^2 + 5(5x) - 4) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-5(2cos(1\u03c0x + 1) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(3sin(3\u03c0(1(-5x^2) + 5) - 3) - 2) + 1)/(3(3sin(3\u03c0(1(-5x^2) + 5) - 3) - 2) + 1) 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) = 1sin(1\u03c0(-1|(-4x^2 + 5) + 1| - 3) - 1) - 3 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) = 1|((-3(3sin(3\u03c0(-5x^2 + 3) + 3) + 2) + 2)/(1(3sin(3\u03c0(-5x^2 + 3) + 3) + 2) + 2)) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1|(3sin(1\u03c0x + 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) = (2(1cos(3\u03c0(-4x^2 + 3x - 5) + 3) + 1) + 1)/(2(1cos(3\u03c0(-4x^2 + 3x - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1sin(2\u03c0((-1x + 3)/(1x + 2))) - 2)^2 + 5(1sin(2\u03c0((-1x + 3)/(1x + 2))) - 2) - 4 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) = 2|(4(3cos(2\u03c0x + 2) - 3) + 5) - 2| - 1 positive?\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(2cos(2\u03c0(-1(-1|x + 3|)^2 - 1(-1|x + 3|) + 4) + 3) + 1) - 2)/(-3(2cos(2\u03c0(-1(-1|x + 3|)^2 - 1(-1|x + 3|) + 4) + 3) + 1) - 2) negative?\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(2(-2cos(1\u03c0x + 2) + 2)^2 - 4(-2cos(1\u03c0x + 2) + 2) - 4) - 2)/(-2(2(-2cos(1\u03c0x + 2) + 2)^2 - 4(-2cos(1\u03c0x + 2) + 2) - 4) - 3) 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((-3(3cos(3\u03c0x - 2)) - 1)/(-2(3cos(3\u03c0x - 2)))) 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(3sin(2\u03c0(1|x + 3| - 1) - 2) + 2))/(-1(3sin(2\u03c0(1|x + 3| - 1) - 2) + 2) + 2) 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) = 1(-2|(5(2sin(1\u03c0x + 2))^2 - 2(2sin(1\u03c0x + 2)) + 4) - 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) = -1(3cos(3\u03c0((-2x - 3)/(2x + 1)) + 2) + 1) - 5 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) = -5((1(-1cos(1\u03c0x - 2)) + 3)/(-1(-1cos(1\u03c0x - 2)) - 1))^2 + 3((1(-1cos(1\u03c0x - 2)) + 3)/(-1(-1cos(1\u03c0x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3cos(3\u03c0(1(-1x^2 - 2x - 2) - 2) + 2) - 3) - 2| + 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) = 3(1sin(3\u03c0(-5((-2x - 2)/(-1x))^2 + 5((-2x - 2)/(-1x)) - 1) - 3) + 1) - 4 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) = -1sin(3\u03c0((1(1|(1x - 5) + 1|) - 3)/(3(1|(1x - 5) + 1|) - 2))) - 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) = 5(3|((1(-1sin(1\u03c0x + 3) + 2) - 3)/(1(-1sin(1\u03c0x + 3) + 2)))| + 1)^2 + 2(3|((1(-1sin(1\u03c0x + 3) + 2) - 3)/(1(-1sin(1\u03c0x + 3) + 2)))| + 1) + 1 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) = 2cos(1\u03c0(-5(-3x - 4)^2 + 2(-3x - 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": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(2|(5x^2 - 2x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(3\u03c0(-4(-1x + 5)^2 - 2(-1x + 5) - 4) + 2) + 1 is strictly decreasing.\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) = (1(3(1cos(2\u03c0(-1x - 3)))^2 - 3(1cos(2\u03c0(-1x - 3))) - 2) - 1)/(-3(3(1cos(2\u03c0(-1x - 3)))^2 - 3(1cos(2\u03c0(-1x - 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) = 3cos(3\u03c0(5(1x^2 - 4x - 5) - 2) - 3) negative?\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) = 5(-4(1sin(3\u03c0(1|x - 1| + 1) - 2) + 2)^2 + 4(1sin(3\u03c0(1|x - 1| + 1) - 2) + 2) - 3) + 5 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) = 4((3(-2|(2sin(1\u03c0x + 1) + 3) - 2| - 1) + 3)/(2(-2|(2sin(1\u03c0x + 1) + 3) - 2| - 1)))^2 - 5((3(-2|(2sin(1\u03c0x + 1) + 3) - 2| - 1) + 3)/(2(-2|(2sin(1\u03c0x + 1) + 3) - 2| - 1))) + 5 is strictly decreasing.\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) = -1sin(1\u03c0(-4(-3|x + 1| + 2)^2 + 5(-3|x + 1| + 2) + 1) - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-5(-1|x - 2| - 1) + 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": "99", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(3cos(2\u03c0(1|(3x^2 - 2x - 2) + 1|) - 1) + 1) - 3)/(-3(3cos(2\u03c0(1|(3x^2 - 2x - 2) + 1|) - 1) + 1) - 3) 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) = -3((1(2|(3sin(1\u03c0x - 1) + 1) + 3| + 1) - 3)/(-2(2|(3sin(1\u03c0x - 1) + 1) + 3| + 1) - 2))^2 + 2((1(2|(3sin(1\u03c0x - 1) + 1) + 3| + 1) - 3)/(-2(2|(3sin(1\u03c0x - 1) + 1) + 3| + 1) - 2)) - 2 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) = -2sin(3\u03c0(-3((-1x + 1)/(-1x + 2))^2 + 4((-1x + 1)/(-1x + 2))) - 2) 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((-2(3cos(1\u03c0(3|x + 3| + 2))) - 1)/(3(3cos(1\u03c0(3|x + 3| + 2))) + 2))^2 + 1((-2(3cos(1\u03c0(3|x + 3| + 2))) - 1)/(3(3cos(1\u03c0(3|x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2cos(2\u03c0(1|(5x + 2) + 2| - 1) + 2) - 1)^2 + 3(-2cos(2\u03c0(1|(5x + 2) + 2| - 1) + 2) - 1) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(3((-1(3sin(2\u03c0x + 2) - 3) + 3)/(-1(3sin(2\u03c0x + 2) - 3) - 3)) - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-2(-3(1cos(2\u03c0x - 2) - 3)^2 - 2(1cos(2\u03c0x - 2) - 3) - 4) + 5) + 3)/(2(-2(-3(1cos(2\u03c0x - 2) - 3)^2 - 2(1cos(2\u03c0x - 2) - 3) - 4) + 5) - 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) = -1((-1(1sin(2\u03c0x - 1) - 1) - 2)/(3(1sin(2\u03c0x - 1) - 1) - 2)) + 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) = (-3(1sin(2\u03c0(2|x| - 1) - 2) - 2) + 1)/(1(1sin(2\u03c0(2|x| - 1) - 2) - 2) + 2) 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|((-3(2cos(1\u03c0x - 3) - 2) - 1)/(-1(2cos(1\u03c0x - 3) - 2) - 3)) + 2| + 1)^2 - 3(2|((-3(2cos(1\u03c0x - 3) - 2) - 1)/(-1(2cos(1\u03c0x - 3) - 2) - 3)) + 2| + 1) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((-3(1cos(3\u03c0x + 1) + 1) - 3)/(3(1cos(3\u03c0x + 1) + 1) + 2)) - 1 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) = (2(-2(1|(1cos(1\u03c0x + 1) - 3) + 3|) + 1) - 1)/(1(-2(1|(1cos(1\u03c0x + 1) - 3) + 3|) + 1)) 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) = 1sin(2\u03c0(2|(5(3x^2 + 4x + 5) + 5)| + 1) - 1) + 2 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(-2cos(1\u03c0(2(-2|x - 2| + 2) + 2) - 1) - 2) - 2)/(-3(-2cos(1\u03c0(2(-2|x - 2| + 2) + 2) - 1) - 2) - 1) 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) = 2(-2|(3(2cos(3\u03c0x + 1) + 2)^2 - 4(2cos(3\u03c0x + 1) + 2) - 1)| - 3) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3sin(2\u03c0(-1|x| + 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) = -2sin(3\u03c0((3(-2|x + 3| + 3) + 2)/(-3(-2|x + 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": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-2|(1(2sin(2\u03c0x + 2) + 3)^2 + 3(2sin(2\u03c0x + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2|(3cos(2\u03c0x - 1) - 3) - 3| - 2)^2 + 4(2|(3cos(2\u03c0x - 1) - 3) - 3| - 2) + 3 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(2|(-2sin(1\u03c0((2x - 3)/(2x)) + 1)) - 3| + 1) + 5 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) = 3|(-1(-1cos(3\u03c0x + 1) + 2) + 5) - 3| + 1 positive?\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(-5(3cos(2\u03c0x - 1) + 3)^2 - 2(3cos(2\u03c0x - 1) + 3) + 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) = -4(-2|((-2(-3cos(2\u03c0x + 2) + 1))/(1(-3cos(2\u03c0x + 2) + 1) - 2))| - 1) - 4 positive?\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|(-3cos(1\u03c0(4(4x + 2)^2 - 3(4x + 2) - 4) + 1) - 2) + 3| + 1 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) = -3(-1sin(3\u03c0(1|(2x + 5) - 3| - 1) - 2) - 1)^2 + 2(-1sin(3\u03c0(1|(2x + 5) - 3| - 1) - 2) - 1) - 5 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) = -1cos(2\u03c0(-4(-1x - 4)^2 - 3(-1x - 4) + 3) - 3) - 1 is strictly increasing.\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) = -3(-4((-1(-1cos(2\u03c0x - 2) + 1))/(-2(-1cos(2\u03c0x - 2) + 1) - 1))^2 - 3((-1(-1cos(2\u03c0x - 2) + 1))/(-2(-1cos(2\u03c0x - 2) + 1) - 1)) - 3) - 5 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|(-2sin(2\u03c0x - 3) - 3) + 2| - 3) + 3)/(2(-1|(-2sin(2\u03c0x - 3) - 3) + 2| - 3) + 1) 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) = -1sin(1\u03c0((1(-4x + 1) + 3)/(1(-4x + 1) - 3)) - 1) + 3 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) = 1|(-3cos(3\u03c0(2x + 2)) + 2) + 2| + 1 is strictly decreasing.\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) = 2sin(1\u03c0(-2|(3(-4x - 1)^2 + 5(-4x - 1) + 1) + 3| + 1) - 3) + 3 negative?\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) = -4(1sin(3\u03c0(3x^2 - 3x - 3) + 1) - 2) + 1 positive?\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) = 1(3|(2cos(2\u03c0x - 2) - 1) + 3| + 3) + 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) = 1(-2cos(3\u03c0(-1((1x + 1)/(2x + 3))^2 - 1((1x + 1)/(2x + 3)) + 5) - 1)) + 5 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) = -3sin(1\u03c0(-3|((-1x + 3)/(2x - 1))| - 1) + 1) is strictly increasing.\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) = (3(1(-3sin(2\u03c0x - 3) - 3) + 3) - 1)/(3(1(-3sin(2\u03c0x - 3) - 3) + 3) - 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) = -2(2((2(-3sin(2\u03c0x + 2) - 1) - 3)/(2(-3sin(2\u03c0x + 2) - 1) + 1)))^2 + 5(2((2(-3sin(2\u03c0x + 2) - 1) - 3)/(2(-3sin(2\u03c0x + 2) - 1) + 1))) + 1 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) = 3(-3(-1cos(3\u03c0x + 1) + 2) + 1)^2 + 2(-3(-1cos(3\u03c0x + 1) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3sin(1\u03c0(-3(3|x + 1| - 2) - 5) - 3) + 1) - 1)/(-2(3sin(1\u03c0(-3(3|x + 1| - 2) - 5) - 3) + 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3|(-1(-1cos(2\u03c0x - 3) + 1)^2 + 2(-1cos(2\u03c0x - 3) + 1) + 2) + 3| + 1) - 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) = -1sin(1\u03c0(4(-2|x| + 2)^2 + 4(-2|x| + 2) - 4) + 2) is strictly decreasing.\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|(-1(1cos(3\u03c0x + 2) - 1) - 1) - 1| + 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) = -5(1cos(3\u03c0(2|x - 1| + 1)) + 3) - 3 is strictly decreasing.\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(-2cos(2\u03c0(-2(1|x - 1| + 2)^2 - 2(1|x - 1| + 2) + 2) + 2)) + 3)/(3(-2cos(2\u03c0(-2(1|x - 1| + 2)^2 - 2(1|x - 1| + 2) + 2) + 2)) - 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) = 3cos(3\u03c0(1(1|x - 2| - 1) + 2) + 1) - 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) = 1(1sin(2\u03c0(-5(3|x - 2| + 2) - 5)) - 2)^2 - 1(1sin(2\u03c0(-5(3|x - 2| + 2) - 5)) - 2) 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) = 2sin(1\u03c0(-2(3|((-2x + 3)/(2x - 1)) + 2| + 1) - 1) - 3) - 1 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) = -3cos(1\u03c0(-1|((3(2x + 1) - 1)/(-1(2x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2(2cos(3\u03c0(1|x| + 2) - 3) + 1) - 3)^2 + 5(-2(2cos(3\u03c0(1|x| + 2) - 3) + 1) - 3) - 5 is strictly decreasing.\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(5(-1|x - 1| + 3)^2) - 3) - 1 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(-2cos(1\u03c0(-3|x - 1| + 3) - 1) - 2)^2 - 5(-2cos(1\u03c0(-3|x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-2(-2sin(3\u03c0x - 1) + 2) - 2)| + 1 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) = (1(1|(5(2sin(3\u03c0x - 2)) + 2) - 3| + 3))/(-3(1|(5(2sin(3\u03c0x - 2)) + 2) - 3| + 3) + 1) is strictly increasing.\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(-2sin(2\u03c0(2(-3x - 1)^2 + 5(-3x - 1) - 2) + 2) + 3))/(-3(-2sin(2\u03c0(2(-3x - 1)^2 + 5(-3x - 1) - 2) + 2) + 3) + 1) 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) = 1cos(1\u03c0(4(1|x + 1|)) + 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(-2cos(2\u03c0(-3x^2 - 3x - 5) + 1) - 3) - 2)/(1(-2cos(2\u03c0(-3x^2 - 3x - 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": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2sin(2\u03c0((-1x - 3)/(-3x + 1)) + 3) - 1)^2 - 2(2sin(2\u03c0((-1x - 3)/(-3x + 1)) + 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": "For the function f(x) = -1((-1(1|(-1sin(2\u03c0x - 3) - 2) - 2| - 1) - 1)/(2(1|(-1sin(2\u03c0x - 3) - 2) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(-3(2|(-4x^2 - 3x + 1) + 3| - 3) - 5) - 1) + 2 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|(-1(-1cos(1\u03c0x + 1) + 2)^2 + 1(-1cos(1\u03c0x + 1) + 2) - 2) + 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) = 1(1sin(2\u03c0(1(-2|x + 3| + 3) - 3) + 2) - 1)^2 + 5(1sin(2\u03c0(1(-2|x + 3| + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(4(3sin(3\u03c0x + 3) - 3)^2 + 1(3sin(3\u03c0x + 3) - 3) - 5) - 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) = 1|(-1(-2sin(2\u03c0x - 3)) + 5) + 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) = -1|(2cos(2\u03c0(-1x + 3) - 1) - 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) = -4((3(-2|(-1cos(1\u03c0x - 2) + 1) + 2| + 1) - 3)/(2(-2|(-1cos(1\u03c0x - 2) + 1) + 2| + 1) - 1))^2 - 4((3(-2|(-1cos(1\u03c0x - 2) + 1) + 2| + 1) - 3)/(2(-2|(-1cos(1\u03c0x - 2) + 1) + 2| + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(-3(3|(-4x^2 + 5x - 5) + 3| + 2) + 4)) is strictly increasing.\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) = (-1(2(1sin(2\u03c0x + 3) + 2) + 1) + 1)/(-3(2(1sin(2\u03c0x + 3) + 2) + 1) + 4) 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(-2(3(-2sin(1\u03c0x - 3) - 1)^2 + 3(-2sin(1\u03c0x - 3) - 1)) - 5) + 3)/(3(-2(3(-2sin(1\u03c0x - 3) - 1)^2 + 3(-2sin(1\u03c0x - 3) - 1)) - 5) + 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) = -1sin(2\u03c0(5(-1|(-3x^2 + 4x + 5)| - 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": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1|(-1sin(2\u03c0x + 1) - 2) - 1| - 3)^2 - 5(-1|(-1sin(2\u03c0x + 1) - 2) - 1| - 3) + 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) = (1(3|(-2cos(3\u03c0(-4x - 1)) - 2) + 1| - 1) - 3)/(2(3|(-2cos(3\u03c0(-4x - 1)) - 2) + 1| - 1) + 3) 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) = -1|(-3sin(3\u03c0(-1(4x^2 - 1x + 4) + 5) + 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": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(4(2cos(2\u03c0x + 2)) + 3)^2 + 3(4(2cos(2\u03c0x + 2)) + 3) - 3 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) = (3(1(3cos(1\u03c0(4x + 2) - 1) + 1)^2 - 3(3cos(1\u03c0(4x + 2) - 1) + 1) - 5) - 3)/(1(1(3cos(1\u03c0(4x + 2) - 1) + 1)^2 - 3(3cos(1\u03c0(4x + 2) - 1) + 1) - 5) + 2) is strictly decreasing.\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) = 3|(-1(1cos(3\u03c0x + 2) - 2) - 1) + 2| + 3 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) = -1sin(2\u03c0(4(5x + 2)^2 + 4(5x + 2) - 4) - 1) + 2 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|(2(2(3sin(2\u03c0x - 1) - 1) - 4)^2 - 1(2(3sin(2\u03c0x - 1) - 1) - 4) + 5) - 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) = (1(3cos(1\u03c0(3(2x^2 - 1x)) + 3) - 3) - 2)/(3(3cos(1\u03c0(3(2x^2 - 1x)) + 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": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2(-2sin(2\u03c0x - 2))^2 - 2(-2sin(2\u03c0x - 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(-1cos(1\u03c0(-2|x + 1| - 3) - 1) + 2)^2 - 3(-1cos(1\u03c0(-2|x + 1| - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(2\u03c0(-5(3x^2 + 4x + 3) - 1)) + 1 positive?\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) = -3|(1(3sin(1\u03c0x - 3) + 2)^2 - 4(3sin(1\u03c0x - 3) + 2) - 5) + 3| + 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) = -1cos(2\u03c0(3|(2x - 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": "77", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-4(-4(1sin(1\u03c0x - 3) - 1) - 4)^2 + 1(-4(1sin(1\u03c0x - 3) - 1) - 4) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1cos(2\u03c0(4x) + 3) - 2) + 3)/(-3(-1cos(2\u03c0(4x) + 3) - 2)) 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) = -1(1(-2|(1sin(3\u03c0x + 3) - 1)| + 3))^2 - 4(1(-2|(1sin(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1sin(3\u03c0(-5x) - 1) - 2) - 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1cos(1\u03c0(3|(-1x^2 - 3x)| - 1) + 1) + 1) + 2)/(1(1cos(1\u03c0(3|(-1x^2 - 3x)| - 1) + 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": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(2(3cos(1\u03c0x) + 3) + 1) - 2)/(-2(2(3cos(1\u03c0x) + 3) + 1) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2cos(3\u03c0(2((2x + 2)/(3x - 2))) + 2) - 2)^2 + 2(2cos(3\u03c0(2((2x + 2)/(3x - 2))) + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(2\u03c0(-1((-2(1x + 5) - 1)/(1(1x + 5) + 2))^2 + 1((-2(1x + 5) - 1)/(1(1x + 5) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(5(2sin(2\u03c0(-2x + 5) + 2))^2 + 1(2sin(2\u03c0(-2x + 5) + 2)) - 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": "78", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((1(2cos(3\u03c0x - 1) - 1) + 3)/(-3(2cos(3\u03c0x - 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": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1((1(-1cos(1\u03c0x - 2) - 1) - 1)/(2(-1cos(1\u03c0x - 2) - 1) + 1))^2 + 4((1(-1cos(1\u03c0x - 2) - 1) - 1)/(2(-1cos(1\u03c0x - 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": "29", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3cos(2\u03c0(-1(2|x - 2| - 2) + 5) - 2) - 3)^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) = -4(-1(-2cos(1\u03c0(-3|x + 2| + 1) - 1) - 3)^2 - 3(-2cos(1\u03c0(-3|x + 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": "45", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(4(3sin(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0((-3(-1|x| - 2) + 3)/(-1(-1|x| - 2) - 1)) - 1) + 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) = -3cos(2\u03c0(1|((-1x - 2)/(-2x + 3)) - 1| + 1)) + 1 positive?\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) = (2(-2|(-5(1sin(2\u03c0x + 1) - 1)^2 - 5(1sin(2\u03c0x + 1) - 1) - 5) - 2| + 2) + 2)/(-2(-2|(-5(1sin(2\u03c0x + 1) - 1)^2 - 5(1sin(2\u03c0x + 1) - 1) - 5) - 2| + 2) + 3) is strictly increasing.\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(2\u03c0(-3((-2(-5x + 4) + 3)/(-1(-5x + 4) - 3))^2 - 5) - 3) + 3 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|(-1sin(3\u03c0((-1x)/(-1x + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0((3(1x + 1) - 1)/(-1(1x + 1) - 2)) + 1) positive?\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) = -3sin(2\u03c0(-3(3((-2x + 2)/(3x))^2 + 3((-2x + 2)/(3x)) + 3) + 2) - 3) - 1 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) = 2cos(1\u03c0(-5((-3x - 1)/(2x + 3))^2 - 3((-3x - 1)/(2x + 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": "For the function f(x) = (2(-2cos(3\u03c0(-3x^2 + 5x - 3) + 3)) + 2)/(2(-2cos(3\u03c0(-3x^2 + 5x - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3((3(-2cos(2\u03c0x + 2) - 2) + 1)/(1(-2cos(2\u03c0x + 2) - 2) + 1)) - 2) + 1| - 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) = 2|((3(-1cos(2\u03c0(-4x^2 + 5x) + 1) + 1) + 3)/(-3(-1cos(2\u03c0(-4x^2 + 5x) + 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": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3sin(3\u03c0(1|(1x^2 - 1x - 5) - 1| + 2)) - 3))/(-2(3sin(3\u03c0(1|(1x^2 - 1x - 5) - 1| + 2)) - 3) + 3) positive?\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(1\u03c0(-2|(-3x^2 - 3x)| - 3) + 3) + 3 negative?\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(2|(-2(1cos(3\u03c0x - 3) - 1)^2 + 1(1cos(3\u03c0x - 3) - 1) - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-3|(2sin(1\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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1sin(1\u03c0(3|x + 3| + 2) + 2) + 2) - 4 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) = -1(5(1cos(1\u03c0x - 1) - 3)^2 + 4(1cos(1\u03c0x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(3\u03c0(5((-1(4x - 4) - 3)/(-1(4x - 4)))^2 + 1((-1(4x - 4) - 3)/(-1(4x - 4))) - 2) + 3) + 1 positive?\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|(-2cos(3\u03c0((1(-3x^2 + 2x + 2))/(-1(-3x^2 + 2x + 2) - 2)) - 3) + 3)| + 1 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) = 1cos(2\u03c0(-1(-2x^2 - 3x - 1) + 4) - 3) + 3 is strictly decreasing.\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) = -3(-2sin(1\u03c0(1x - 2) + 3) + 3)^2 + 5(-2sin(1\u03c0(1x - 2) + 3) + 3) - 2 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) = 3(-3|(2sin(1\u03c0x) + 1) - 1| - 2)^2 - 2(-3|(2sin(1\u03c0x) + 1) - 1| - 2) - 4 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(-2cos(2\u03c0(5x^2 + 3x - 1) + 2) + 2) - 2)/(-3(-2cos(2\u03c0(5x^2 + 3x - 1) + 2) + 2) + 1) positive?\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) = (2(-3|(2cos(3\u03c0x + 1) - 2) - 1| + 1) + 1)/(-3(-3|(2cos(3\u03c0x + 1) - 2) - 1| + 1) - 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) = 1sin(3\u03c0(4((1x - 3)/(-2x)) - 3) + 1) - 1 negative?\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) = 4(-4((-1(-3cos(3\u03c0x - 3) - 3) + 3)/(1(-3cos(3\u03c0x - 3) - 3) + 3)) + 2)^2 - 4(-4((-1(-3cos(3\u03c0x - 3) - 3) + 3)/(1(-3cos(3\u03c0x - 3) - 3) + 3)) + 2) + 1 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) = (-2(3cos(1\u03c0(3x + 2) - 2) + 1) - 1)/(-2(3cos(1\u03c0(3x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(2|(3cos(3\u03c0(1x)) + 3) - 2| - 2) + 3)/(-2(2|(3cos(3\u03c0(1x)) + 3) - 2| - 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3(-1sin(3\u03c0x + 2) + 2)^2 + 1(-1sin(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-1(-1|x + 1| - 1)^2 - 5(-1|x + 1| - 1) - 3) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(3(-1|x + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3sin(1\u03c0(1|((-1x - 2)/(-1x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(2(-1cos(2\u03c0(2|x + 2|)) - 2)^2 + 4(-1cos(2\u03c0(2|x + 2|)) - 2) + 1) + 3)/(-2(2(-1cos(2\u03c0(2|x + 2|)) - 2)^2 + 4(-1cos(2\u03c0(2|x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3((1(3sin(3\u03c0x) - 2) - 3)/(2(3sin(3\u03c0x) - 2) + 3))^2 + 5((1(3sin(3\u03c0x) - 2) - 3)/(2(3sin(3\u03c0x) - 2) + 3)) - 3) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}