{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|((-3(-2sin(1\u03c0x - 2) + 2) + 2)/(2(-2sin(1\u03c0x - 2) + 2) + 1)) - 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) = 1sin(1\u03c0(3(3x^2 - 2x + 5)^2 + 1(3x^2 - 2x + 5) - 2) + 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0(-1|(3|x + 3|)| + 2) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-1(-5x^2 + 1)^2 - 4(-5x^2 + 1) - 2)^2 - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-3(-3sin(1\u03c0x + 3) - 3) + 5)^2 + 5(-3(-3sin(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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-4(-2sin(1\u03c0x - 2) + 1) - 4)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(3|(1sin(1\u03c0x - 1) + 1) - 1|) + 1)/(3(3|(1sin(1\u03c0x - 1) + 1) - 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) = -3sin(3\u03c0(-5(-2x - 4)) - 2) - 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) = -1cos(1\u03c0(-1(4x^2 + 3x - 5)^2 - 5(4x^2 + 3x - 5) + 4) + 1) - 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(5(1cos(3\u03c0x + 1) - 3)^2 + 3(1cos(3\u03c0x + 1) - 3) - 1) - 3)/(-2(5(1cos(3\u03c0x + 1) - 3)^2 + 3(1cos(3\u03c0x + 1) - 3) - 1) - 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) = -1(2cos(2\u03c0((2x - 1)/(-2x + 3)) + 1)) - 5 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) = 3cos(2\u03c0(-2(-2|x - 3|) - 2) - 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(2\u03c0(-1|((-3x - 3)/(1x)) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0((1(-3|x - 1| + 3) + 2)/(2(-3|x - 1| + 3) + 2)) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(2cos(2\u03c0(1x^2 + 4x + 4))) + 5 is strictly increasing.\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) = -3(-1cos(2\u03c0(-1x^2 - 2x - 1) + 1) - 3)^2 - 3(-1cos(2\u03c0(-1x^2 - 2x - 1) + 1) - 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2|(-3cos(2\u03c0x - 1) + 2) - 3|) - 2)/(3(2|(-3cos(2\u03c0x - 1) + 2) - 3|)) 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((-3x + 2)/(-1x + 3)) - 1))^2 + 2(-1cos(1\u03c0((-3x + 2)/(-1x + 3)) - 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(-2((3x - 3)/(1x - 2))^2 - 3((3x - 3)/(1x - 2)) + 2) + 1) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1sin(1\u03c0(-4x^2 - 1x + 5) + 3) + 1)^2 + 3 is strictly decreasing.\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) = -3sin(1\u03c0(3((-1x + 2)/(1x + 3))^2 + 4((-1x + 2)/(1x + 3)) - 1) - 1) - 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) = -3|(-1(-3x^2 - 3x + 4)^2 - 4(-3x^2 - 3x + 4) - 2) - 1| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1cos(1\u03c0(-3x^2 - 2x - 3) - 2) - 3) + 2)/(1(1cos(1\u03c0(-3x^2 - 2x - 3) - 2) - 3)) 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) = 3|(-2(2cos(2\u03c0x - 3) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2|(-1sin(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) = 3cos(2\u03c0(-1(2x^2 - 1x + 3) + 1)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(1(3|x + 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": "90", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2|(3sin(1\u03c0x + 3) + 3) - 1|)^2 - 3(-2|(3sin(1\u03c0x + 3) + 3) - 1|) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(-5(-3|x + 3| - 1)^2 - 1(-3|x + 3| - 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": "For the function f(x) = 1sin(2\u03c0(4((1x + 3)/(-1x))^2 - 3((1x + 3)/(-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": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-5(1sin(2\u03c0x + 2) - 1) + 5)^2 - 1(-5(1sin(2\u03c0x + 2) - 1) + 5) - 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) = -1(-3(1sin(3\u03c0x - 1) + 3)^2 - 1(1sin(3\u03c0x - 1) + 3) + 2)^2 - 4 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) = (-1(1(3sin(3\u03c0x - 1) - 3) - 5) + 1)/(-3(1(3sin(3\u03c0x - 1) - 3) - 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(1sin(1\u03c0(-4x^2 + 3x - 2) + 3) + 3) + 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(1\u03c0(-4((1x - 1)/(3x - 2)) + 5) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1cos(2\u03c0(4x^2 - 5x + 5)) + 2)^2 - 5(1cos(2\u03c0(4x^2 - 5x + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-3(-3cos(1\u03c0x - 1) + 1)^2 - 4(-3cos(1\u03c0x - 1) + 1) - 3) + 1)/(2(-3(-3cos(1\u03c0x - 1) + 1)^2 - 4(-3cos(1\u03c0x - 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) = -1|(1(2cos(3\u03c0x - 3) - 1)^2 + 5) + 1| - 2 negative?\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) = 1sin(1\u03c0(-1|(-3|x - 2| + 2) - 3| - 2) + 1) + 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) = (-3(1|(3cos(3\u03c0x - 3) - 1) + 1| - 1) + 1)/(1(1|(3cos(3\u03c0x - 3) - 1) + 1| - 1) - 2) is strictly increasing.\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(-2(-2cos(1\u03c0x) + 1)^2 + 3(-2cos(1\u03c0x) + 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((-1(3cos(2\u03c0x + 1) - 3) + 1)/(-3(3cos(2\u03c0x + 1) - 3) - 1)) + 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(-4(1cos(1\u03c0x - 3))^2 - 4)^2 + 5(-4(1cos(1\u03c0x - 3))^2 - 4) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3cos(2\u03c0(1x^2 + 2x - 1) + 1) + 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-1(-1x + 5) - 3) + 3) 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(2|x| - 1) - 3) - 1)^2 - 2 is strictly increasing.\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) = -1sin(2\u03c0(-2|(-5x^2 - 3x - 4) - 3| + 2)) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(3(-5x)^2 + 4(-5x)) - 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2sin(2\u03c0(-4x + 3) - 2) - 2) + 2)/(-2(-2sin(2\u03c0(-4x + 3) - 2) - 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0((1(-1|x| + 3))/(-2(-1|x| + 3) + 3)) + 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1(1cos(2\u03c0x - 3) - 3) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((2(3cos(3\u03c0x + 2) - 3) - 2)/(1(3cos(3\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|(-2(-2cos(2\u03c0x + 3) - 1)^2 + 2(-2cos(2\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-3((-1x + 2)/(-2x + 3))^2 - 5((-1x + 2)/(-2x + 3)) + 3) + 2) 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) = 3(5(2cos(1\u03c0x + 3) - 3)^2 + 4(2cos(1\u03c0x + 3) - 3) - 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-5(-2sin(3\u03c0x + 3) - 1)^2 - 5(-2sin(3\u03c0x + 3) - 1) - 1)^2 + 4(-5(-2sin(3\u03c0x + 3) - 1)^2 - 5(-2sin(3\u03c0x + 3) - 1) - 1) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3|(-1x^2 + 4x + 4) + 1| + 2)^2 - 5(-3|(-1x^2 + 4x + 4) + 1| + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1(-1sin(2\u03c0x) + 3)^2 + 1(-1sin(2\u03c0x) + 3) - 4) - 2| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((-3(1sin(1\u03c0x - 2) - 2) - 3)/(1(1sin(1\u03c0x - 2) - 2) - 3)) + 4 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|(3cos(1\u03c0(3x - 2) - 2) + 1) + 2| + 3 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) = 3cos(2\u03c0(-4(-2|x + 1| + 2) + 2) - 3) + 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) = -2sin(3\u03c0((1((2x + 3)/(-1x + 2)) + 2)/(-2((2x + 3)/(-1x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2|(-3cos(3\u03c0x + 3) - 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) = 2(1sin(2\u03c0(1x^2 + 4x - 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": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2sin(1\u03c0(5x^2 - 2x - 4) - 3) - 2))/(-1(2sin(1\u03c0(5x^2 - 2x - 4) - 3) - 2) - 3) is strictly decreasing.\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((-1(1cos(1\u03c0x + 2) - 2) + 2)/(3(1cos(1\u03c0x + 2) - 2) + 3))^2 + 5((-1(1cos(1\u03c0x + 2) - 2) + 2)/(3(1cos(1\u03c0x + 2) - 2) + 3)) + 5 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) = 4(3(-3cos(1\u03c0x - 3) - 1) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((-2(3cos(1\u03c0x + 2) + 1) + 1)/(1(3cos(1\u03c0x + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2((-1(3cos(2\u03c0x) - 1) - 1)/(-1(3cos(2\u03c0x) - 1) + 2)) - 2)/(1((-1(3cos(2\u03c0x) - 1) - 1)/(-1(3cos(2\u03c0x) - 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": "For the function f(x) = 1sin(3\u03c0(3(-2x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-2cos(3\u03c0((-2x + 2)/(2x)) + 1) - 2))/(-3(-2cos(3\u03c0((-2x + 2)/(2x)) + 1) - 2) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2|(1sin(2\u03c0x + 1) + 1) + 1| + 2) + 1| + 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|(2sin(1\u03c0((-1x + 3)/(2x - 1)) - 2) - 2) + 1| + 3 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) = 2|(-1(2cos(2\u03c0x))^2 - 4(2cos(2\u03c0x)) + 2) - 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-3(-5x - 4) + 2) - 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "97", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-4(1x^2 + 5x - 4)^2 - 5(1x^2 + 5x - 4) - 2) + 2| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1cos(3\u03c0(5x^2 - 3x + 1)) + 3)^2 - 4(1cos(3\u03c0(5x^2 - 3x + 1)) + 3) - 4 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) = -4(2|(2cos(3\u03c0x + 1) + 3) - 2|) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0(1(-3x + 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": "90", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3\u03c0(-1|((2x - 3)/(-2x)) - 1| + 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) = 1(-2sin(1\u03c0(-5x - 5)) - 1) - 3 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) = -1sin(1\u03c0(2(1x - 4)^2 - 1(1x - 4) - 1) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1(2cos(3\u03c0x - 2))^2 - 2(2cos(3\u03c0x - 2)) + 3)^2 - 3(1(2cos(3\u03c0x - 2))^2 - 2(2cos(3\u03c0x - 2)) + 3) - 5 negative?\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(3cos(2\u03c0(-2x^2 - 1x - 1) + 3)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(-3(-3x - 4)^2 + 2(-3x - 4) + 2) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(2sin(2\u03c0(1x^2 + 3x - 5) - 1) + 3)^2 - 5(2sin(2\u03c0(1x^2 + 3x - 5) - 1) + 3) + 1 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(-3sin(1\u03c0(-3x^2 - 4x - 5) - 2) - 2) 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) = 1(-5(3sin(3\u03c0x))^2 + 1(3sin(3\u03c0x)) + 5)^2 - 3(-5(3sin(3\u03c0x))^2 + 1(3sin(3\u03c0x)) + 5) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-3|(3cos(3\u03c0x - 2) - 2) + 2| + 2) + 1| + 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(-2cos(2\u03c0x - 3) - 3) - 2) - 2)/(-3(-5(-2cos(2\u03c0x - 3) - 3) - 2) + 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) = 5(2cos(1\u03c0((-2x - 1)/(-1x - 2)) + 3) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0(-1(-4x) - 4) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1cos(3\u03c0(-3|x + 1| - 2) - 3) + 1) + 1| + 1 is strictly decreasing.\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) = 1|(-3cos(2\u03c0((1x)/(-2x - 3)) - 2) - 2) - 2| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1(-2sin(1\u03c0x - 1) - 3) - 4) - 3| + 2 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) = -5(4(3sin(3\u03c0x) + 3)^2 - 2(3sin(3\u03c0x) + 3) + 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(2|(2x^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": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(2\u03c0(-2((-3x)/(-3x - 1))^2 - 1((-3x)/(-3x - 1))) + 1) 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(2(1sin(2\u03c0x - 2) + 3)^2 + 4(1sin(2\u03c0x - 2) + 3) + 3) - 1)/(-2(2(1sin(2\u03c0x - 2) + 3)^2 + 4(1sin(2\u03c0x - 2) + 3) + 3) + 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) = 2cos(1\u03c0(4(-3|x + 2| + 3)^2 - 2(-3|x + 2| + 3) - 2) + 2) - 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) = -2|(5(3cos(2\u03c0x - 1) + 2) - 1) + 2| + 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) = 2cos(2\u03c0(3(-3x - 1) - 3) + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(2sin(2\u03c0(-2|x + 2| + 1))) + 3| + 1 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(-2sin(1\u03c0(4x + 3) - 1) + 2)^2 - 5(-2sin(1\u03c0(4x + 3) - 1) + 2) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2sin(3\u03c0((-1x)/(2x - 3)) - 1) + 1)^2 - 3(2sin(3\u03c0((-1x)/(2x - 3)) - 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": "For the function f(x) = 1cos(1\u03c0((-2(2x^2 - 2x + 4) + 3)/(-1(2x^2 - 2x + 4) + 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) = (-3(-2(-2sin(3\u03c0x) - 1) - 5) + 2)/(-3(-2(-2sin(3\u03c0x) - 1) - 5) - 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) = 5((-3(-2cos(1\u03c0x + 3) + 3) - 3)/(3(-2cos(1\u03c0x + 3) + 3) - 2)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3\u03c0(4((3x - 2)/(-3x)) + 2) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((1(-2sin(2\u03c0x - 1) - 3) - 2)/(1(-2sin(2\u03c0x - 1) - 3) + 2))^2 + 1((1(-2sin(2\u03c0x - 1) - 3) - 2)/(1(-2sin(2\u03c0x - 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": "For the function f(x) = -1(2|(1sin(3\u03c0x - 2) + 2)| - 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(1(1cos(3\u03c0x + 2) + 2) + 5) - 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) = 3|(-3sin(1\u03c0(-5x + 5) - 1) + 3) + 2| - 1 negative?\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) = (2(-2(-1sin(3\u03c0x + 1) + 1) - 5) - 1)/(-2(-2(-1sin(3\u03c0x + 1) + 1) - 5)) 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((-2(1sin(3\u03c0x - 3) - 1) + 2)/(2(1sin(3\u03c0x - 3) - 1) + 2))^2 + 1((-2(1sin(3\u03c0x - 3) - 1) + 2)/(2(1sin(3\u03c0x - 3) - 1) + 2)) + 5 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(4(-2sin(1\u03c0x - 2) + 2) - 5) + 3)/(1(4(-2sin(1\u03c0x - 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) = 2cos(1\u03c0(1(4x + 2)^2 - 3(4x + 2) + 5) - 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) = (-1(-2sin(2\u03c0(2x^2 - 5) - 2) - 3) - 1)/(-2(-2sin(2\u03c0(2x^2 - 5) - 2) - 3) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-3|(-1cos(3\u03c0x + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(2\u03c0((2((1x + 2)/(3x - 2)) - 2)/(1((1x + 2)/(3x - 2)) + 2)) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(3sin(1\u03c0(3x + 3) - 3)) - 2)/(1(3sin(1\u03c0(3x + 3) - 3)) + 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) = 1cos(2\u03c0(-1(2x^2 - 2x - 4) + 4) - 1) - 2 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) = 2sin(1\u03c0(1|((3x - 1)/(3x + 1)) - 2| + 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1|(-3cos(1\u03c0x - 1) - 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) = -3|(-2(2cos(2\u03c0x) + 1) + 3) - 2| + 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) = 3(-1cos(2\u03c0(3x^2 - 4x + 3) - 3) + 3)^2 - 1(-1cos(2\u03c0(3x^2 - 4x + 3) - 3) + 3) + 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|(-3sin(2\u03c0(1x + 4) - 1) + 1) + 1| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(2\u03c0(3((2x - 2)/(3x + 2))^2 + 3((2x - 2)/(3x + 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": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(1|((3x - 1)/(3x)) - 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": "For the function f(x) = -3cos(3\u03c0(-1|(3x^2 - 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": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2cos(2\u03c0(-5x^2 + 3x) + 1)) - 2 is strictly increasing.\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) = 1sin(2\u03c0(2|(-1|x + 3| + 2) + 1| + 2) - 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-3sin(1\u03c0(1x - 2) + 1) + 3)^2 + 4(-3sin(1\u03c0(1x - 2) + 1) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3sin(3\u03c0(5x^2 + 4x) - 3) - 3) + 2)/(-2(3sin(3\u03c0(5x^2 + 4x) - 3) - 3) + 3) positive?\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(2\u03c0x + 1) + 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) = 1(-4(3sin(3\u03c0x - 2) - 1)^2 + 5(3sin(3\u03c0x - 2) - 1) + 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3sin(2\u03c0((-3x - 3)/(1x + 2)) + 2)) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1\u03c0(3(3x + 3)^2 + 4(3x + 3) + 4) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-3cos(3\u03c0((-2x - 1)/(3x - 1))) + 1) + 2)/(-3(-3cos(3\u03c0((-2x - 1)/(3x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3cos(3\u03c0(4x^2 + 5x - 1) + 3) + 1)^2 - 4(-3cos(3\u03c0(4x^2 + 5x - 1) + 3) + 1) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-3(3cos(3\u03c0x - 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": "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|x - 2| - 1) - 3| + 2) - 3) + 2 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) = -1cos(1\u03c0(2((2x + 3)/(-3x - 1))^2 + 2((2x + 3)/(-3x - 1)) - 5) + 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-3sin(2\u03c0((3x + 2)/(1x)) + 2) - 3)| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(-3|(3x - 5) + 2| + 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) = (2(1cos(3\u03c0(-3|x - 2|) + 1) + 3) + 2)/(-2(1cos(3\u03c0(-3|x - 2|) + 1) + 3)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2((1(-1cos(1\u03c0x + 2) - 2) + 1)/(2(-1cos(1\u03c0x + 2) - 2) + 3))^2 - 3((1(-1cos(1\u03c0x + 2) - 2) + 1)/(2(-1cos(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0((2(2x^2 + 4x + 2) - 1)/(3(2x^2 + 4x + 2)))) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3\u03c0(1(3|x - 2| - 3) - 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0((-1(-5x + 5) - 3)/(-2(-5x + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0((1((-2x - 2)/(2x - 1)) + 2)/(2((-2x - 2)/(2x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-4(2cos(1\u03c0x - 1))^2 + 2(2cos(1\u03c0x - 1)) - 2)| + 3 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(1sin(1\u03c0(-5x^2 + 4x + 3) - 2) + 1) + 1)/(-1(1sin(1\u03c0(-5x^2 + 4x + 3) - 2) + 1) - 1) is strictly increasing.\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|(-2sin(3\u03c0(-3|x|) + 1) + 1) - 3| + 2 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) = (-1(3cos(3\u03c0(3x^2 - 4x + 1) + 3) - 3) + 1)/(2(3cos(3\u03c0(3x^2 - 4x + 1) + 3) - 3) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(1|(1x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(1(2cos(1\u03c0x - 2) + 2) + 4) - 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) = (2(-2|(3sin(2\u03c0x + 3)) + 1| + 3) - 2)/(-2(-2|(3sin(2\u03c0x + 3)) + 1| + 3) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-3(2sin(2\u03c0x - 2) - 3)^2 + 5(2sin(2\u03c0x - 2) - 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) = 4(-2(1cos(2\u03c0x) - 1)^2 - 3(1cos(2\u03c0x) - 1) + 1)^2 - 5(-2(1cos(2\u03c0x) - 1)^2 - 3(1cos(2\u03c0x) - 1) + 1) + 4 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) = 1(1cos(3\u03c0(2|x - 1|) - 1)) - 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) = -3cos(2\u03c0(-1(1|x + 3| - 1)^2 - 3(1|x + 3| - 1) + 2) - 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3(3sin(2\u03c0x - 3))^2 + 5(3sin(2\u03c0x - 3)) + 5) - 3)/(2(-3(3sin(2\u03c0x - 3))^2 + 5(3sin(2\u03c0x - 3)) + 5)) 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) = 2(2(-3cos(3\u03c0x + 2) - 1)^2 - 4(-3cos(3\u03c0x + 2) - 1) + 5)^2 - 2(2(-3cos(3\u03c0x + 2) - 1)^2 - 4(-3cos(3\u03c0x + 2) - 1) + 5) - 3 positive?\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) = -5(-2cos(1\u03c0(-2x + 5) - 2) + 2)^2 - 1(-2cos(1\u03c0(-2x + 5) - 2) + 2) - 4 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(2sin(2\u03c0(3|x| - 2) + 1) - 2) - 2 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(-1sin(1\u03c0(-3|x - 2| - 1)) - 1) + 1)/(3(-1sin(1\u03c0(-3|x - 2| - 1)) - 1) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0(1(-2x^2 + 1x - 2)^2 - 2(-2x^2 + 1x - 2) - 4)) - 2 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) = -1cos(3\u03c0((-1(2x + 2) + 3)/(-1(2x + 2) - 2)) - 2) + 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) = -5(2(3sin(1\u03c0x + 1)) - 1)^2 + 1(2(3sin(1\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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(1\u03c0(-3(-3x^2 - 1x - 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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3|(-2cos(1\u03c0x + 1) + 2) - 2| - 1) + 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-3cos(2\u03c0(-4x - 3) - 1) - 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(-3|(1x^2 - 5x - 2) + 3| + 3) + 1) - 3 negative?\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) = 3(-2sin(2\u03c0(1|x + 2| + 2) - 2) + 3)^2 - 5(-2sin(2\u03c0(1|x + 2| + 2) - 2) + 3) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-4(-2sin(2\u03c0x + 1) + 1) - 4) + 3)/(1(-4(-2sin(2\u03c0x + 1) + 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": "For the function f(x) = -5(3|(1cos(3\u03c0x - 1) + 3) - 1| + 1)^2 - 5(3|(1cos(3\u03c0x - 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": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-3(2|x + 3| - 3)^2 + 3(2|x + 3| - 3) + 5) - 3| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3((3(3sin(3\u03c0x + 1) - 3) + 2)/(-1(3sin(3\u03c0x + 1) - 3))))/(-1((3(3sin(3\u03c0x + 1) - 3) + 2)/(-1(3sin(3\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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((-2(-1cos(3\u03c0x - 3) + 3) + 2)/(2(-1cos(3\u03c0x - 3) + 3) - 1))^2 + 3((-2(-1cos(3\u03c0x - 3) + 3) + 2)/(2(-1cos(3\u03c0x - 3) + 3) - 1)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3cos(2\u03c0(-2x^2 - 3x + 5) + 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": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2sin(2\u03c0(1x + 4) + 3) + 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) = 1sin(3\u03c0(4(2|x + 1| + 3) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0(3|(2x) + 1| - 3) - 3) + 3 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) = 3(3|(-1cos(3\u03c0x + 2) - 1) - 2| - 3)^2 - 2(3|(-1cos(3\u03c0x + 2) - 1) - 2| - 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1cos(3\u03c0(-5x - 5) + 1) + 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3cos(2\u03c0((-1x)/(3x - 2)) - 1) - 2))/(-2(-3cos(2\u03c0((-1x)/(3x - 2)) - 1) - 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2\u03c0(-2(2|x| + 2) + 1) + 2) + 1 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) = -2sin(2\u03c0(2(1|x + 2| + 2)^2 + 1(1|x + 2| + 2) - 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": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-3cos(1\u03c0(3|x + 1| - 2)) + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(1cos(2\u03c0((2x)/(1x + 1)) - 3) + 3) - 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(-3(-3cos(2\u03c0x - 1) - 2) + 5) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2sin(2\u03c0(-5x - 3)) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "109", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(3cos(3\u03c0(5x^2 - 2x + 1) + 3) + 3)^2 - 2(3cos(3\u03c0(5x^2 - 2x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(2cos(2\u03c0((-3x - 2)/(-2x + 3)))) - 3| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(5(-2cos(3\u03c0x - 2) + 1))^2 + 5(5(-2cos(3\u03c0x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(1\u03c0(1|(-4x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-3sin(3\u03c0((3x - 3)/(-1x)) - 2) - 3)^2 - 2(-3sin(3\u03c0((3x - 3)/(-1x)) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(1sin(1\u03c0(3|x - 1| - 3) - 2) + 2) + 3)/(-2(1sin(1\u03c0(3|x - 1| - 3) - 2) + 2) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-1cos(2\u03c0((-2x + 3)/(3x)) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((2(3sin(1\u03c0x - 2) + 3))/(1(3sin(1\u03c0x - 2) + 3) - 3))^2 - 3((2(3sin(1\u03c0x - 2) + 3))/(1(3sin(1\u03c0x - 2) + 3) - 3)) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-5(3|x - 2| - 2)^2 - 5(3|x - 2| - 2) + 3) + 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-5(3sin(2\u03c0x - 3) - 3) + 1)^2 + 2(-5(3sin(2\u03c0x - 3) - 3) + 1) - 2 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) = 4(3sin(1\u03c0(-5x - 1) - 2) + 3)^2 + 2(3sin(1\u03c0(-5x - 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": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(-5(1x^2 - 3x + 1) + 3) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(1\u03c0(-1(-1x^2 - 4x - 3) + 1) - 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) = -4(2(-3cos(2\u03c0x - 3) - 1)^2 + 3(-3cos(2\u03c0x - 3) - 1) + 1)^2 + 2(2(-3cos(2\u03c0x - 3) - 1)^2 + 3(-3cos(2\u03c0x - 3) - 1) + 1) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3sin(1\u03c0((-2x + 3)/(1x + 2)) - 1) + 1) + 3| + 3 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) = -1sin(3\u03c0(3|(-1x^2 + 4x - 1) - 2|) - 2) + 1 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(3((3x - 3)/(3x - 1))^2 - 2((3x - 3)/(3x - 1)) - 4) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(1\u03c0(-3(-4x^2 - 5x + 5)^2 + 1(-4x^2 - 5x + 5) + 5) - 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) = -3(2cos(1\u03c0((1x + 3)/(1x)) - 3) - 2)^2 - 2(2cos(1\u03c0((1x + 3)/(1x)) - 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((1(3sin(1\u03c0x + 3) + 1))/(-2(3sin(1\u03c0x + 3) + 1) + 3)) + 3)/(-2((1(3sin(1\u03c0x + 3) + 1))/(-2(3sin(1\u03c0x + 3) + 1) + 3)) - 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) = 1sin(2\u03c0(-2(2x^2 + 3x)^2 + 4(2x^2 + 3x) - 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) = -5(-3cos(1\u03c0(3|x + 1| - 2) - 3) - 1)^2 + 3(-3cos(1\u03c0(3|x + 1| - 2) - 3) - 1) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(-1|(-2|x| - 3) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((1(-3sin(1\u03c0x - 2) - 3) - 2)/(-2(-3sin(1\u03c0x - 2) - 3) - 1)) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2(2cos(1\u03c0x + 2) + 2)^2 + 2(2cos(1\u03c0x + 2) + 2) + 1)^2 - 4(-2(2cos(1\u03c0x + 2) + 2)^2 + 2(2cos(1\u03c0x + 2) + 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(2sin(1\u03c0((1x + 2)/(-1x - 3)) + 2) - 1)^2 + 2(2sin(1\u03c0((1x + 2)/(-1x - 3)) + 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(2sin(3\u03c0(-5x^2 - 2) - 3) - 2) - 3| + 3 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) = (1(2sin(1\u03c0((3x - 1)/(-1x + 2)) + 1) - 2))/(-2(2sin(1\u03c0((3x - 1)/(-1x + 2)) + 1) - 2) - 2) 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) = 2sin(1\u03c0(2((-2x + 1)/(-3x))^2 - 1((-2x + 1)/(-3x)) - 3) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2((3(2cos(1\u03c0x - 1) + 3))/(-3(2cos(1\u03c0x - 1) + 3) - 1)) - 2)/(2((3(2cos(1\u03c0x - 1) + 3))/(-3(2cos(1\u03c0x - 1) + 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) = 2|(2(3sin(3\u03c0x - 1) + 1)^2 + 5(3sin(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": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2sin(1\u03c0((3x - 1)/(2x + 2)) + 2) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-2cos(1\u03c0((-2x - 2)/(-3x + 2)) - 1) + 3) + 2)/(1(-2cos(1\u03c0((-2x - 2)/(-3x + 2)) - 1) + 3) - 3) 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) = 1cos(3\u03c0(-5(-4x^2 - 2x + 3) + 1) + 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(2cos(2\u03c0(-3|x| + 3) + 2) + 3) - 5 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(4(2|x| - 1)^2 - 2(2|x| - 1) - 3)^2 + 4(4(2|x| - 1)^2 - 2(2|x| - 1) - 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2sin(2\u03c0(-4x^2 + 1x - 5)) + 3) + 2| + 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) = -1sin(1\u03c0(-4(2|x - 1| - 2) + 4) + 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(-1(1x^2 + 1x)^2 + 3(1x^2 + 1x) + 5) - 2) - 3 negative?\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) = (2(-1cos(3\u03c0(3|x - 2| - 1) - 2)))/(-1(-1cos(3\u03c0(3|x - 2| - 1) - 2)) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0((2((1x + 2)/(1x - 2)) + 2)/(-2((1x + 2)/(1x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(3((-3x)/(2x + 2))^2 - 2((-3x)/(2x + 2)) - 5) + 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) = 5((1(-3cos(3\u03c0x + 3) + 1) + 2)/(3(-3cos(3\u03c0x + 3) + 1) + 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) = -3(3sin(2\u03c0(-3|x - 1|) - 3) - 3)^2 + 4(3sin(2\u03c0(-3|x - 1|) - 3) - 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1|(4x^2 + 1x - 4) + 1| - 1)|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0((-1(-2|x - 1| + 2) + 1)/(3(-2|x - 1| + 2) + 3)) - 1) + 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) = -3cos(3\u03c0(1(2|x + 2| - 2)^2 + 4(2|x + 2| - 2) - 1) + 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-3|(3cos(1\u03c0x + 2) - 3) - 2| - 2)^2 - 3(-3|(3cos(1\u03c0x + 2) - 3) - 2| - 2) + 4 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(-1(-5x^2 + 1x + 3)^2 + 5(-5x^2 + 1x + 3) + 3) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(-1|(3|x + 2| + 1) + 3|) - 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) = -3|(1|(-3cos(2\u03c0x - 3) + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(2\u03c0(-2|((1x - 3)/(3x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0((-1(1x - 5) - 2)/(-2(1x - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0((3(4x - 5) + 1)/(1(4x - 5))) - 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-5(3x^2 - 2x - 1) + 5) - 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2sin(3\u03c0(-5x^2 + 5x) - 1) - 1)^2 + 3(2sin(3\u03c0(-5x^2 + 5x) - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-5(2sin(1\u03c0x + 2) - 1) - 1) + 2| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3cos(1\u03c0(-4x^2 - 5x + 5) - 2) + 3) - 1)/(-2(-3cos(1\u03c0(-4x^2 - 5x + 5) - 2) + 3) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0(-5(5x^2 + 3x - 4) - 5) - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0(2|(2|x + 2| - 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": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0((-3(-1x) + 1)/(-1(-1x) - 3)) - 3) + 1 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|(1|(-4x^2 - 4x + 4) + 2| - 3) + 2| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-1|(-2sin(1\u03c0x - 3) - 1) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(3|(-3|x - 1| + 3)| - 2)) + 2 is strictly increasing.\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) = -2cos(3\u03c0((1(-3x^2 - 2x + 1) - 3)/(-2(-3x^2 - 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) = -2|(3(-3sin(1\u03c0x - 1) - 2)^2 - 4(-3sin(1\u03c0x - 1) - 2) - 4) - 1| - 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) = 2cos(2\u03c0(-2(-1x^2 + 1x + 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": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2(-3sin(3\u03c0x - 3) + 2)^2 - 2(-3sin(3\u03c0x - 3) + 2) + 5) - 3 is strictly increasing.\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) = -4(-1cos(3\u03c0(-1|x + 2| - 3) - 1) - 1) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0(1(-1x + 5) - 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(-2|(-3|x + 2| + 2)| + 2)^2 - 3(-2|(-3|x + 2| + 2)| + 2) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(2|(3cos(3\u03c0x + 2) + 3)| - 3) + 3)/(-2(2|(3cos(3\u03c0x + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2sin(3\u03c0(4x^2 + 1x - 5) - 2)) - 2| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2((1(-2sin(3\u03c0x + 3) + 3) - 1)/(-2(-2sin(3\u03c0x + 3) + 3) + 3)) + 2)/(2((1(-2sin(3\u03c0x + 3) + 3) - 1)/(-2(-2sin(3\u03c0x + 3) + 3) + 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) = 3|(2|(-2cos(1\u03c0x - 3) - 2) - 3| - 1) - 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0(4((-3x - 1)/(-1x + 2)) + 1) - 2) - 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) = 1cos(2\u03c0((-3(5x^2 + 2x + 5) + 1)/(1(5x^2 + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-3(2|x + 3| - 3)^2 + 3(2|x + 3| - 3) + 4) - 3| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1sin(1\u03c0(1|x + 1|)) + 3) + 2 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) = 4((-1(-1cos(2\u03c0x + 1)) - 1)/(-2(-1cos(2\u03c0x + 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) = -1|(5(-1cos(1\u03c0x - 3)) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2cos(3\u03c0((-2x + 1)/(1x + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1|(-1sin(1\u03c0x + 3) + 3) + 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": "For the function f(x) = 1cos(3\u03c0(2|((-2x - 2)/(3x + 2)) + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-1sin(3\u03c0(1x^2 + 2x - 3) - 1)) + 3)/(2(-1sin(3\u03c0(1x^2 + 2x - 3) - 1))), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-2sin(3\u03c0(2|x|) + 1) + 1)^2 - 2(-2sin(3\u03c0(2|x|) + 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": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-5(1|x - 1| - 1) - 2) + 3) - 1 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|(-2cos(3\u03c0((-1x - 2)/(1x)) - 3) - 3) - 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-4(2sin(3\u03c0x) - 1) - 2) - 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(2sin(2\u03c0((-3x - 2)/(2x)) - 3) - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-3|(-2sin(2\u03c0x)) + 2| - 2) + 2)/(-2(-3|(-2sin(2\u03c0x)) + 2| - 2) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-5(-1x^2 + 4x - 2)^2 + 4) - 3| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(2\u03c0(3(-1x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(2|(3cos(3\u03c0x) - 3)| - 1) - 3)/(-1(2|(3cos(3\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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-5(-2cos(2\u03c0x + 3) + 1) + 5) + 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) = -1|((3(2sin(3\u03c0x) + 2))/(-1(2sin(3\u03c0x) + 2) + 3)) - 2| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3sin(2\u03c0((1x + 1)/(1x)) - 1) - 2) - 1)/(-1(-3sin(2\u03c0((1x + 1)/(1x)) - 1) - 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3cos(1\u03c0((-2x + 2)/(-1x - 2)) + 1) + 3)^2 - 3(-3cos(1\u03c0((-2x + 2)/(-1x - 2)) + 1) + 3) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2sin(3\u03c0(-1x^2 - 1x + 4) - 2) + 3)^2 - 1(2sin(3\u03c0(-1x^2 - 1x + 4) - 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": "24", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(5(5x^2 + 2x + 2)^2 + 1(5x^2 + 2x + 2) + 4)) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-4(-1cos(1\u03c0x + 2))^2 - 3(-1cos(1\u03c0x + 2)) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1(2cos(1\u03c0x - 3) - 1) + 3)^2 + 1(-1(2cos(1\u03c0x - 3) - 1) + 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-3|(-2cos(3\u03c0x - 1) + 3) + 3| - 3)^2 + 1(-3|(-2cos(3\u03c0x - 1) + 3) + 3| - 3) - 5 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((3(-2sin(3\u03c0x + 3) - 1) - 1)/(-2(-2sin(3\u03c0x + 3) - 1) - 1))^2 + 2((3(-2sin(3\u03c0x + 3) - 1) - 1)/(-2(-2sin(3\u03c0x + 3) - 1) - 1)) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3sin(1\u03c0(1x - 3) - 2) + 1) - 3)/(-2(3sin(1\u03c0(1x - 3) - 2) + 1) - 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) = 1(-1cos(3\u03c0(2|x - 3| + 1) + 3) + 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(4(1cos(2\u03c0x + 1) - 1)^2 + 5(1cos(2\u03c0x + 1) - 1) - 3)^2 - 1(4(1cos(2\u03c0x + 1) - 1)^2 + 5(1cos(2\u03c0x + 1) - 1) - 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2|(1sin(1\u03c0x - 3) - 1) - 1| - 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(3((1x - 3)/(-2x + 3))^2 + 4((1x - 3)/(-2x + 3)) + 5) - 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(1|(2|x + 1| - 1) - 3| - 3)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1|(1cos(3\u03c0x - 3) + 1) + 2| - 3) + 5 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) = -2|(2cos(2\u03c0(4x^2 - 1x - 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": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(3|(-3cos(2\u03c0x) + 1) - 2| - 3)^2 + 2(3|(-3cos(2\u03c0x) + 1) - 2| - 3) - 4 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) = -1|(1|(2sin(2\u03c0x - 1) - 3) - 1| - 1) - 3| + 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) = -3cos(3\u03c0(-2|((1x - 1)/(-1x + 2)) + 1| + 2) - 1) + 3 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) = (3(2sin(3\u03c0(-1x + 4) - 2) + 1) - 2)/(-2(2sin(3\u03c0(-1x + 4) - 2) + 1) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(-3((2x - 2)/(-2x + 3))^2 - 2((2x - 2)/(-2x + 3)) - 3) - 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1sin(2\u03c0(3|x + 2| - 1) + 1) + 3) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1sin(2\u03c0(-5x - 2) - 3) - 1))/(-2(-1sin(2\u03c0(-5x - 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": "89", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(3cos(1\u03c0(2|x - 3|) - 2) - 1)^2 + 4(3cos(1\u03c0(2|x - 3|) - 2) - 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) = -5(2(-2cos(3\u03c0x + 3) + 1) + 1)^2 + 2(2(-2cos(3\u03c0x + 3) + 1) + 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) = -2|(-1cos(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3|(-1sin(3\u03c0x + 1) + 1) - 3| - 2) + 3)/(-3(3|(-1sin(3\u03c0x + 1) + 1) - 3| - 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-3(1x + 5)^2 - 1(1x + 5) - 1) - 2) - 2 is strictly increasing.\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) = -4(1(-2cos(2\u03c0x + 1) - 2) + 4)^2 - 4(1(-2cos(2\u03c0x + 1) - 2) + 4) + 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|(-2|(-1sin(3\u03c0x + 1)) - 3| - 2) - 3| + 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) = 3(2cos(3\u03c0(-3|x - 2| - 3)) + 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(-1|(-3x^2 + 4x - 5) + 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": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-1cos(3\u03c0(4x + 3) - 1)) + 5 is strictly increasing.\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) = -1(1cos(3\u03c0(-5x + 2) - 2) - 2)^2 - 3(1cos(3\u03c0(-5x + 2) - 2) - 2) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(1\u03c0(-3|(-2|x - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0(-2|(3x + 4) - 1|) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1|(1cos(2\u03c0x + 2) - 1) - 3| + 1)^2 - 1 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) = 1sin(2\u03c0(-1|(-2|x + 1| + 1) + 1| + 3) - 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(2cos(3\u03c0(-2x - 2) - 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": "79", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1((3(2cos(3\u03c0x) - 3) + 2)/(2(2cos(3\u03c0x) - 3) - 1)) + 1)/(1((3(2cos(3\u03c0x) - 3) + 2)/(2(2cos(3\u03c0x) - 3) - 1))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1sin(1\u03c0((-2x + 2)/(-2x - 3)) + 3) + 2) + 1| - 1 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(-3|(3cos(2\u03c0x + 1) - 1) - 3| - 3) - 3)/(-2(-3|(3cos(2\u03c0x + 1) - 1) - 3| - 3) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-2sin(2\u03c0(3|x - 3| - 2) - 1) - 3)| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(-3(3x - 1))) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-2cos(1\u03c0(-2|x - 3| + 2) - 2) + 2) - 1)/(1(-2cos(1\u03c0(-2|x - 3| + 2) - 2) + 2) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3(3cos(2\u03c0x + 1) + 3)^2 - 5(3cos(2\u03c0x + 1) + 3) + 1) - 3| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-3sin(2\u03c0(-3|x - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(1cos(3\u03c0((-2x - 3)/(-3x + 2))) + 3) - 2)/(2(1cos(3\u03c0((-2x - 3)/(-3x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(-1(4x^2 + 4x - 4)^2 + 2(4x^2 + 4x - 4) - 5) - 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(5(1x^2 - 3x - 4)^2 - 4) + 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(1\u03c0(2((2x + 3)/(2x + 2))^2 - 3((2x + 3)/(2x + 2))) - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(4(-1sin(1\u03c0x + 3) + 1)^2 - 4) + 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) = -1cos(3\u03c0(4(4x) - 3) - 2) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2sin(3\u03c0((2x - 3)/(-1x - 3)) + 3) + 3)^2 + 1(-2sin(3\u03c0((2x - 3)/(-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": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(3\u03c0(1(1|x - 2| - 1) + 5) + 3) + 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) = -1sin(2\u03c0(-2(-3x^2 - 4x - 4)^2 + 1(-3x^2 - 4x - 4) + 4) + 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-1cos(1\u03c0(4x^2 - 3x) - 2) - 3)^2 - 2(-1cos(1\u03c0(4x^2 - 3x) - 2) - 3) - 5 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(2cos(1\u03c0(3|x + 2| + 3) + 1) + 1)^2 - 5(2cos(1\u03c0(3|x + 2| + 3) + 1) + 1) + 5 is strictly decreasing.\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|((2(-2cos(1\u03c0x - 1) + 2) - 2)/(1(-2cos(1\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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(1(2sin(3\u03c0x + 3) - 2) - 3) + 2)/(-1(1(2sin(3\u03c0x + 3) - 2) - 3) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0((2((1x - 2)/(3x + 2)) + 3)/(1((1x - 2)/(3x + 2)))) + 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1\u03c0(4((-1x + 3)/(3x + 3)) + 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) = -2sin(3\u03c0(-1((-3x + 3)/(3x - 2))^2) - 2) - 2 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) = -2cos(1\u03c0(1|((-2x + 1)/(1x + 2)) - 2| + 3) + 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1|(2sin(1\u03c0x + 3) - 3) - 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) = 4(-3(-1cos(1\u03c0x - 1) - 3) - 3) - 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(-5(2cos(1\u03c0x - 1) + 1) - 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(4((-2x - 1)/(-3x - 2)) - 2) + 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-3sin(3\u03c0((-2x + 2)/(-3x - 2)) - 3) - 3) + 2| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0(-2((-1x - 2)/(1x + 1)) - 4) - 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1(1cos(2\u03c0x + 1) + 2) - 2) - 3| - 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(-2cos(3\u03c0(1x^2 + 5x - 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": "36", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2cos(2\u03c0(-3|x|) + 3) + 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(3cos(3\u03c0(-2x + 3) - 1) - 2))/(-2(3cos(3\u03c0(-2x + 3) - 1) - 2) - 1) negative?\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(1(5x + 5) - 4) + 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(2\u03c0(3(-2|x + 2| - 3)^2 - 3(-2|x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0(2|((-1x)/(3x + 2))| - 3)) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(-2|(-2|x + 2| + 1) + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(2\u03c0(5x + 4) + 1) - 1))/(-2(-1sin(2\u03c0(5x + 4) + 1) - 1) + 3) is strictly increasing.\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) = 2|(2(-1cos(1\u03c0x + 2) + 1) - 4) - 2| - 2 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) = 5(-1(2cos(2\u03c0x - 2) + 3) - 5) - 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) = 1((-2(-3sin(2\u03c0x - 1) + 2) + 2)/(1(-3sin(2\u03c0x - 1) + 2))), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((-3(3cos(3\u03c0x + 2) + 2) - 1)/(1(3cos(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": "68", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(1(-1x^2 + 4x)^2 - 4(-1x^2 + 4x) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-2|(2sin(1\u03c0x - 3) - 2) + 2| + 2)^2 + 3(-2|(2sin(1\u03c0x - 3) - 2) + 2| + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3|(2sin(1\u03c0x - 2) + 1) - 2| + 3)^2 + 5(3|(2sin(1\u03c0x - 2) + 1) - 2| + 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) = -2cos(1\u03c0(2|(-2|x - 2| - 1)| - 2) + 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-3(-3sin(1\u03c0x - 3) + 2) - 4) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(-4(4x^2 - 4x - 4) - 1) + 1) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(-3((-3x)/(-2x + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(1\u03c0(2|(-2x - 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": "For the function f(x) = 2(-1cos(1\u03c0(2|x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((-1(2cos(2\u03c0x - 3) + 1) - 1)/(3(2cos(2\u03c0x - 3) + 1) + 1))| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3|(3cos(3\u03c0x - 3) + 2) - 1| + 2) - 1| + 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) = -2cos(2\u03c0((-2(-2|x| + 1))/(3(-2|x| + 1) + 3)) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0((3(-1|x - 2|) - 3)/(-2(-1|x - 2|)))) - 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) = -2|(-1|(2sin(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(1(-2cos(2\u03c0x) - 3)^2 - 4(-2cos(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) = 1|(3cos(1\u03c0(-1x^2 - 2x + 4) - 1) - 2) + 1| + 1 negative?\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) = -5((-2(2cos(3\u03c0x - 3) - 3) - 1)/(-1(2cos(3\u03c0x - 3) - 3) - 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) = 3sin(3\u03c0(1|(2x^2 - 4x - 1) + 1| + 1) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(3(-2cos(3\u03c0x + 3) + 2)^2 - 4(-2cos(3\u03c0x + 3) + 2) + 2) - 2 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) = -1sin(3\u03c0((2(-3x - 3) - 3)/(-1(-3x - 3) + 3)) - 1) + 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) = 3cos(2\u03c0(-3|(-2x) + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((1(1cos(2\u03c0x + 1) + 1) + 1)/(-2(1cos(2\u03c0x + 1) + 1))) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0((-1(-1|x + 3| + 3) - 1)/(1(-1|x + 3| + 3) + 2)) - 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(3\u03c0(1|(-2|x + 3| + 3) + 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": "62", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-2sin(2\u03c0((1x + 3)/(2x + 1)) + 3) + 2) - 3)/(-1(-2sin(2\u03c0((1x + 3)/(2x + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2((3(-3cos(3\u03c0x + 3) - 3) + 3)/(2(-3cos(3\u03c0x + 3) - 3) - 1)) - 1)/(1((3(-3cos(3\u03c0x + 3) - 3) + 3)/(2(-3cos(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2\u03c0(-1(2|x + 2| + 2)^2 + 1(2|x + 2| + 2) - 3) + 3) is strictly increasing.\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) = -3(-1sin(2\u03c0(-1|x + 3| - 3) - 1) - 3)^2 + 4(-1sin(2\u03c0(-1|x + 3| - 3) - 1) - 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3cos(3\u03c0(3x + 4) + 1) - 1) + 2)/(-1(3cos(3\u03c0(3x + 4) + 1) - 1) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(3(4x^2 - 2x + 2) - 2) - 2) + 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) = -3cos(2\u03c0((-2(-2|x| - 2) + 2)/(2(-2|x| - 2) + 3)) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3sin(1\u03c0(-2|x| + 3))) - 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) = (-2(-1sin(1\u03c0((2x + 1)/(-3x + 2)) - 2) + 3) - 2)/(2(-1sin(1\u03c0((2x + 1)/(-3x + 2)) - 2) + 3) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-2|(-2|x - 2|) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(4(-1cos(3\u03c0x + 1) + 2) - 2)^2 + 1(4(-1cos(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(1|(1|x - 3| + 1) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0(-5(3|x - 3| + 2) - 4) + 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) = -3cos(3\u03c0(-5((3x + 1)/(2x - 1))^2 + 5((3x + 1)/(2x - 1)) + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(3cos(3\u03c0(1x - 3)) + 2) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1sin(3\u03c0(-4x + 1) - 3) + 1)^2 - 5(-1sin(3\u03c0(-4x + 1) - 3) + 1) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(1sin(3\u03c0(-3x + 1) - 1) + 2) + 1)/(1(1sin(3\u03c0(-3x + 1) - 1) + 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3cos(2\u03c0(2x - 5) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3(1cos(2\u03c0x + 1) + 1) + 3)^2 - 2(3(1cos(2\u03c0x + 1) + 1) + 3) - 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) = -1(3cos(3\u03c0(2x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0(3((3x + 3)/(-1x - 2))^2 - 1((3x + 3)/(-1x - 2)) + 4) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-3|(-2sin(1\u03c0x) + 3) - 2| - 3)^2 + 4(-3|(-2sin(1\u03c0x) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1cos(1\u03c0(3x^2 - 3x + 4) - 2) + 2) 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) = -4(-1sin(2\u03c0(-4x^2 - 2x + 2) - 1))^2 - 4(-1sin(2\u03c0(-4x^2 - 2x + 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": "For the function f(x) = 3(-3sin(1\u03c0(1x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4((1(-1sin(1\u03c0x + 3)) + 1)/(-2(-1sin(1\u03c0x + 3))))^2 - 4((1(-1sin(1\u03c0x + 3)) + 1)/(-2(-1sin(1\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": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-2cos(2\u03c0(3x^2 + 2x - 3) + 3))^2 + 5(-2cos(2\u03c0(3x^2 + 2x - 3) + 3)) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-1cos(3\u03c0(1|x - 1| - 2) + 2) + 1)^2 + 3(-1cos(3\u03c0(1|x - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(1cos(1\u03c0((-2x + 2)/(1x + 1)) + 1) + 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1(-3cos(3\u03c0x) + 1))^2 + 5(1(-3cos(3\u03c0x) + 1)) - 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) = (-1(3cos(2\u03c0(-2|x - 1| + 3) + 3) + 1) + 2)/(-3(3cos(2\u03c0(-2|x - 1| + 3) + 3) + 1) + 4) 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) = -2cos(1\u03c0(-1(1|x - 2| + 1)^2 - 4(1|x - 2| + 1) - 2)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-3sin(1\u03c0(-5x + 1) + 3))^2 + 3(-3sin(1\u03c0(-5x + 1) + 3)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(3cos(1\u03c0(1x + 3) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1sin(2\u03c0(-2|x + 2| + 2) + 2) - 2) + 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) = -2|(3|(-2cos(3\u03c0x + 1) + 1) - 2| - 1)| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(2|(-1x^2 + 4x - 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": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(3cos(3\u03c0(5x + 3) - 3) + 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-5(1sin(1\u03c0x - 1) + 3)^2 - 4(1sin(1\u03c0x - 1) + 3))^2 + 1(-5(1sin(1\u03c0x - 1) + 3)^2 - 4(1sin(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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(1cos(1\u03c0(4x^2 + 3x + 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": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1sin(1\u03c0(3|x + 1| + 3) - 2) - 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) = 2sin(2\u03c0(3|(-2|x - 1| - 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": "79", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-2sin(3\u03c0(1|x| - 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": "For the function f(x) = 1sin(3\u03c0(2|(5x) + 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": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(1|((-1x + 3)/(1x)) - 3| + 2) + 2) - 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(2sin(3\u03c0(1|x - 2| + 3) - 2) - 3) + 3)/(-2(2sin(3\u03c0(1|x - 2| + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-3sin(3\u03c0x - 3) - 2) - 5)^2 - 3(1(-3sin(3\u03c0x - 3) - 2) - 5) - 3 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|(-3|(2sin(3\u03c0x - 2) - 3) - 1| + 3) - 2| - 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) = 5(3sin(1\u03c0(1x + 1) + 3) + 2)^2 - 5(3sin(1\u03c0(1x + 1) + 3) + 2) - 2 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) = -1cos(2\u03c0(2|(-3x^2 - 1x + 3) + 3| - 1) + 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3sin(2\u03c0(2|x + 3| - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(2cos(3\u03c0((-2x + 2)/(2x + 1)) + 1) - 1) + 1)/(-3(2cos(3\u03c0((-2x + 2)/(2x + 1)) + 1) - 1) + 1) 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) = 3(2sin(2\u03c0(2|x - 3| + 2) - 3) + 3) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1((-2(-1sin(2\u03c0x + 1) - 2) + 3)/(-3(-1sin(2\u03c0x + 1) - 2))) + 1)/(1((-2(-1sin(2\u03c0x + 1) - 2) + 3)/(-3(-1sin(2\u03c0x + 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) = (1((-2(2sin(3\u03c0x - 1) + 3) + 1)/(2(2sin(3\u03c0x - 1) + 3))) + 1)/(-2((-2(2sin(3\u03c0x - 1) + 3) + 1)/(2(2sin(3\u03c0x - 1) + 3)))) 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) = 5(1|(-1cos(1\u03c0x + 3)) - 3| + 3)^2 + 5(1|(-1cos(1\u03c0x + 3)) - 3| + 3) + 5 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) = (2((1(1cos(3\u03c0x + 1) - 2) - 3)/(-1(1cos(3\u03c0x + 1) - 2) - 1)) - 2)/(3((1(1cos(3\u03c0x + 1) - 2) - 3)/(-1(1cos(3\u03c0x + 1) - 2) - 1)) + 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) = 1cos(3\u03c0(-2|(-3|x + 1| - 3) - 3| - 3) - 2) - 1 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) = 1(3|(2cos(2\u03c0x + 3) + 2) - 1| + 1)^2 + 1(3|(2cos(2\u03c0x + 3) + 2) - 1| + 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1|(2cos(3\u03c0x) - 1) - 3| - 1) - 3)/(-2(-1|(2cos(3\u03c0x) - 1) - 3| - 1) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1cos(1\u03c0(2|x| + 3) + 2) + 3)^2 - 4(-1cos(1\u03c0(2|x| + 3) + 2) + 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) = (2(-1(-2sin(3\u03c0x + 3))^2 + 1(-2sin(3\u03c0x + 3))) + 1)/(2(-1(-2sin(3\u03c0x + 3))^2 + 1(-2sin(3\u03c0x + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-1(3cos(1\u03c0x - 2) + 2) + 2)/(2(3cos(1\u03c0x - 2) + 2) + 2)) + 2 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) = 2|(2cos(3\u03c0(-5x + 5) + 3) + 3) - 2| - 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) = 1(3|(1sin(3\u03c0x + 2) - 1) - 1| + 2) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(-5((-3x - 2)/(-2x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(1|(1|x + 1| - 3) - 3| + 1) + 1) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-2cos(2\u03c0(-1x + 3) + 1) + 2) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3\u03c0(-3(3|x - 1|))) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0((-1(-3x^2 + 1x))/(-3(-3x^2 + 1x) - 1)) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5((2(1cos(2\u03c0x - 3)) - 1)/(2(1cos(2\u03c0x - 3)) - 3))^2 - 3((2(1cos(2\u03c0x - 3)) - 1)/(2(1cos(2\u03c0x - 3)) - 3)) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(4((-1x - 3)/(1x + 2)) + 1) - 1) - 2 positive?\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(-3(2sin(2\u03c0x - 2) - 3)^2 + 1(2sin(2\u03c0x - 2) - 3) - 4)^2 + 3(-3(2sin(2\u03c0x - 2) - 3)^2 + 1(2sin(2\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": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-1sin(1\u03c0(-3|x + 3| + 1) + 1) - 2)^2 - 4(-1sin(1\u03c0(-3|x + 3| + 1) + 1) - 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(3\u03c0((3(2|x + 1| - 2) + 3)/(3(2|x + 1| - 2))) - 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) = -4(-2cos(2\u03c0(4x^2 - 1x - 1) + 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": "29", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-3cos(1\u03c0((-2x + 3)/(2x + 3)) - 1) - 1) + 2| - 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) = -1(-1(-2sin(3\u03c0x - 1))^2 + 3(-2sin(3\u03c0x - 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": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((1(2cos(1\u03c0x) - 1) - 1)/(-3(2cos(1\u03c0x) - 1) - 1))^2 - 3((1(2cos(1\u03c0x) - 1) - 1)/(-3(2cos(1\u03c0x) - 1) - 1)) - 2 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) = -2sin(2\u03c0((-3(3|x - 1|) + 1)/(2(3|x - 1|) - 3)) + 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(4(-3x^2 + 2x)) - 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2sin(2\u03c0((-1x - 2)/(3x + 2)) + 3) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3sin(1\u03c0((-3x - 1)/(-1x - 2))) - 3) + 1)/(2(-3sin(1\u03c0((-3x - 1)/(-1x - 2))) - 3) + 3) 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) = 2|(1|(3cos(3\u03c0x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-2(-1cos(2\u03c0x - 1) - 3) - 3) + 1| + 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) = -1sin(1\u03c0(-5(-1x^2 - 3x + 1)^2 - 2(-1x^2 - 3x + 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": "29", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-1sin(3\u03c0(-1|x + 1|) - 2) + 2) - 2| + 3 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) = -1sin(1\u03c0(-5(1x)^2 + 1(1x) - 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": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(1|(-1cos(2\u03c0x + 1) - 3) - 1| - 2) - 4 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) = -3sin(3\u03c0(3(-2|x - 3| + 2) - 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(3(2x - 5)^2 - 5(2x - 5)) + 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) = 4(2sin(3\u03c0(3x^2 + 2x + 4) - 3) + 2)^2 + 1(2sin(3\u03c0(3x^2 + 2x + 4) - 3) + 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(2(3|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": "83", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(4(3cos(2\u03c0x))^2 + 3(3cos(2\u03c0x)) + 1)^2 + 1(4(3cos(2\u03c0x))^2 + 3(3cos(2\u03c0x)) + 1) - 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(-3|(-1sin(1\u03c0x + 1) + 2) - 1| - 1)^2 - 1(-3|(-1sin(1\u03c0x + 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": "For the function f(x) = -3(2cos(2\u03c0((-1x + 3)/(-1x + 2)) + 3))^2 - 4(2cos(2\u03c0((-1x + 3)/(-1x + 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": "8", "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 - 5) + 4) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-2sin(1\u03c0(2x^2 + 3x) - 3) - 3)^2 - 4(-2sin(1\u03c0(2x^2 + 3x) - 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) = (3((3(2sin(1\u03c0x) + 1))/(-1(2sin(1\u03c0x) + 1) - 1)) + 2)/(2((3(2sin(1\u03c0x) + 1))/(-1(2sin(1\u03c0x) + 1) - 1)) + 2) 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) = (3(-2sin(2\u03c0(-3x^2 - 4x - 1)) + 2) - 2)/(-3(-2sin(2\u03c0(-3x^2 - 4x - 1)) + 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(1(-1sin(3\u03c0x - 2) + 1)^2 - 3(-1sin(3\u03c0x - 2) + 1) - 5) + 5 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) = -2sin(3\u03c0((2((-2x + 1)/(-1x - 3)) - 3)/(-3((-2x + 1)/(-1x - 3)) + 1)) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3sin(1\u03c0(-3|x + 3|)) - 3)^2 + 3(-3sin(1\u03c0(-3|x + 3|)) - 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-1(2cos(2\u03c0x + 1) + 3)^2 + 1(2cos(2\u03c0x + 1) + 3) + 2)^2 + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(3(-3x - 3)^2 - 5(-3x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(3\u03c0(-4(3|x - 3| - 2) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-4(-3sin(1\u03c0x + 2) + 2) - 3) - 3)/(-2(-4(-3sin(1\u03c0x + 2) + 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) = 5(-3(-1sin(1\u03c0x + 3) + 1)^2 - 3(-1sin(1\u03c0x + 3) + 1) + 2)^2 + 1(-3(-1sin(1\u03c0x + 3) + 1)^2 - 3(-1sin(1\u03c0x + 3) + 1) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0((-1((-2x - 3)/(1x - 3)) + 3)/(-3((-2x - 3)/(1x - 3)) + 4)) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(2(1sin(2\u03c0x - 2) + 1)^2 - 2(1sin(2\u03c0x - 2) + 1) + 4) + 3)/(1(2(1sin(2\u03c0x - 2) + 1)^2 - 2(1sin(2\u03c0x - 2) + 1) + 4) + 2) 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) = -2sin(1\u03c0(-2|(3|x - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1(-2|x + 1| + 2)^2) - 3| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3\u03c0(-4((-1x + 1)/(1x))^2 - 3((-1x + 1)/(1x)) + 4) - 3) + 3 positive?\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|(-3sin(2\u03c0((2x + 2)/(2x - 1)) - 1) - 2) + 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) = 1(3(-3sin(1\u03c0x - 2) + 3)^2 + 4(-3sin(1\u03c0x - 2) + 3) - 1)^2 + 1(3(-3sin(1\u03c0x - 2) + 3)^2 + 4(-3sin(1\u03c0x - 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": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3((1(-1cos(2\u03c0x) - 3) + 3)/(3(-1cos(2\u03c0x) - 3) + 1))^2 - 1((1(-1cos(2\u03c0x) - 3) + 3)/(3(-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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2cos(1\u03c0(3|x + 2| - 3) - 3) - 3) + 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) = 5(4(2sin(3\u03c0x) + 3) - 5)^2 - 4(4(2sin(3\u03c0x) + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(1\u03c0((-2(2|x + 1| + 3) - 2)/(-1(2|x + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(1sin(3\u03c0(-3|x - 2| - 1) - 2) - 2) - 1)/(-3(1sin(3\u03c0(-3|x - 2| - 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": "85", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0((2(-1|x - 3| + 2) + 1)/(-1(-1|x - 3| + 2))) + 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1sin(3\u03c0(3x^2 - 1) - 3)) - 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) = -2|(-2sin(1\u03c0(2|x| + 3) - 3) - 2) + 3| - 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) = -3(2sin(2\u03c0(-5x - 2) - 3) + 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(1|(1|x + 1| - 2)| - 2)^2 - 5(1|(1|x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(3\u03c0(-1(1|x - 2| - 3) + 1) - 2) + 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) = 2|(3cos(3\u03c0((1x)/(-1x + 2)) + 2) - 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) = -4(2|(3cos(3\u03c0x - 2) - 3) - 1| + 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1cos(2\u03c0(1x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2(-1cos(3\u03c0x - 3) + 3) - 5) + 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) = 2|(4(2cos(3\u03c0x + 1) - 2) + 4)| - 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) = -4((-1(3cos(2\u03c0x - 2) - 1) - 2)/(3(3cos(2\u03c0x - 2) - 1) - 2))^2 + 4((-1(3cos(2\u03c0x - 2) - 1) - 2)/(3(3cos(2\u03c0x - 2) - 1) - 2)) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "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) - 3) - 2| + 3) - 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) = 2cos(1\u03c0((3((-3x + 3)/(-3x + 2)))/(-2((-3x + 3)/(-3x + 2)) + 3)) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-1(2x + 3) - 4) - 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(3sin(1\u03c0(1x - 4) + 2) - 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2sin(2\u03c0(1x^2 + 5x + 2) - 1) + 3) + 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(3sin(3\u03c0(4x + 2) - 3) - 1) - 5 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) = -3cos(2\u03c0((-1(1x - 2) + 3)/(1(1x - 2) + 3)) + 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(-5(-5x + 4)^2 + 4(-5x + 4) - 4) + 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-2|(-3|x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(3((-2x + 2)/(1x + 2))^2 - 5((-2x + 2)/(1x + 2)) + 5) - 1) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((1(3sin(3\u03c0x - 1) + 2) + 3)/(1(3sin(3\u03c0x - 1) + 2)))| + 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|(1|(2|x + 3| - 3) - 1| - 1)| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-3cos(2\u03c0(4x^2 - 4x + 4) - 1) + 1)^2 - 3(-3cos(2\u03c0(4x^2 - 4x + 4) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3sin(2\u03c0(-1|x - 3| - 2) - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(-1|(-2|x - 2| + 2) - 3| - 3) - 3) - 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) = -2sin(3\u03c0(1((-2x + 3)/(-1x + 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) = -4(-3sin(3\u03c0(1x - 5) + 3) + 3)^2 + 1(-3sin(3\u03c0(1x - 5) + 3) + 3) + 4 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) = 3sin(1\u03c0(-3|((2x - 2)/(-2x)) + 3| + 2)) - 3 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) = 3cos(2\u03c0((1(2|x + 1| - 1) - 2)/(3(2|x + 1| - 1) + 1)) - 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((-1(1cos(3\u03c0x + 1) - 1) + 1)/(2(1cos(3\u03c0x + 1) - 1) - 1))^2 + 1((-1(1cos(3\u03c0x + 1) - 1) + 1)/(2(1cos(3\u03c0x + 1) - 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) = -3sin(3\u03c0(-5(-1|x - 2| + 2)^2 - 3(-1|x - 2| + 2) + 3) + 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0(-5(1x^2 + 1x - 5) - 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(2|(-2x^2 - 3x + 4) - 2|) - 3) - 2 is strictly increasing.\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) = 1|(3|(-2sin(3\u03c0x - 1)) + 3| - 3) + 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(3sin(2\u03c0(5x - 3) + 2) - 2) + 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0((2(-1x^2 - 2x + 5) - 1)/(-3(-1x^2 - 2x + 5) + 4)) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3\u03c0(1|(-2x - 4)| + 1) + 1) + 1 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) = -3cos(3\u03c0(-4(-2|x + 1|) - 2)) + 2 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) = -3sin(2\u03c0(-3(-1x^2 + 3) - 4) - 2) + 1 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) = -3cos(2\u03c0(2|(4x - 5) + 2|) - 3) 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(-2cos(2\u03c0((-3x)/(-2x - 3)) + 1) - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(2\u03c0((3(4x - 5))/(1(4x - 5) + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(2sin(3\u03c0x - 2) - 3) + 3| + 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(3sin(1\u03c0(1x^2 + 4x - 4) - 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": "55", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((2(3cos(3\u03c0x + 1)) - 1)/(-1(3cos(3\u03c0x + 1)) - 2)) - 3| + 1 negative?\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(-3(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) - 1) - 3)/(3(-3(-2cos(3\u03c0x - 3) + 3)^2 + 2(-2cos(3\u03c0x - 3) + 3) - 1) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3(-2|x + 2| + 1)^2 - 1)^2 + 5(-3(-2|x + 2| + 1)^2 - 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(1cos(2\u03c0(3|x + 1| + 1) + 1) + 2)^2 - 3(1cos(2\u03c0(3|x + 1| + 1) + 1) + 2) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2\u03c0(3|(-5x^2 + 5x + 3) - 1| + 3) - 2) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(5(1cos(1\u03c0x - 1) - 2)^2 + 5(1cos(1\u03c0x - 1) - 2)) + 3| + 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) = 2cos(1\u03c0(-3(-3x^2 + 2x - 2) - 5) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(2\u03c0(1((3x - 1)/(2x)) + 4) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1sin(3\u03c0(2x - 5) - 3) - 3) - 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(1\u03c0((3(2|x + 3| - 1) + 1)/(3(2|x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0((-1(-1|x - 3| + 3) + 3)/(-2(-1|x - 3| + 3) + 1)) - 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0((-2(-2x - 2) - 2)/(-3(-2x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-3(-1x - 4) + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(1(-2x + 2)^2 - 2(-2x + 2) - 4) + 1) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(2sin(1\u03c0(4x - 1) + 1) - 2) + 1)/(3(2sin(1\u03c0(4x - 1) + 1) - 2) - 2) 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) = -2sin(2\u03c0(-1|(5x + 3) + 1| + 3) - 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-1sin(1\u03c0(1x - 5) - 1) - 2) - 3| - 2 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(2(1|x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(4(-1sin(1\u03c0x - 1) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(3cos(2\u03c0(3|x - 3| + 2) + 2) - 1) + 4 negative?\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(-3|(-1x + 5) + 3| + 2) - 3) - 3 is strictly increasing.\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) = 2(-2sin(1\u03c0(1|x - 1| - 2) - 1) + 3) - 4 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) = -1cos(2\u03c0(-2|((-2x + 1)/(-2x)) + 1|) - 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) = 1cos(1\u03c0(-1|(5x^2 + 4x - 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": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2sin(2\u03c0((-3x - 2)/(-1x + 2)) - 1) - 1) - 4 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) = 4(3(3cos(2\u03c0x + 3) - 2) + 1)^2 - 1(3(3cos(2\u03c0x + 3) - 2) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(5(-3cos(2\u03c0x) + 3)^2 + 5(-3cos(2\u03c0x) + 3) + 1) - 3)/(-1(5(-3cos(2\u03c0x) + 3)^2 + 5(-3cos(2\u03c0x) + 3) + 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) = 3sin(3\u03c0((1(2x - 3) + 3)/(-3(2x - 3) - 3)) - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-3sin(3\u03c0((2x + 3)/(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": "13", "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 + 5x - 5) + 1) - 2) + 3)/(1(-1sin(3\u03c0(4x^2 + 5x - 5) + 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) = -2cos(2\u03c0(-2(1x) - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(3cos(3\u03c0(-3|x + 1| - 1) + 1))| + 1 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) = -1(5(2cos(2\u03c0x + 2) - 2) - 4)^2 - 4(5(2cos(2\u03c0x + 2) - 2) - 4) + 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) = -1cos(2\u03c0(2|((1x)/(1x - 2)) - 3| + 3) + 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(2|(1|x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-5(1sin(2\u03c0x) + 3) - 2)| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(-1|(-1x^2 - 1x - 5) + 1| - 2) - 1) + 1 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(1sin(2\u03c0(-1x^2 - 3x + 4) - 2) - 2)^2 + 4(1sin(2\u03c0(-1x^2 - 3x + 4) - 2) - 2) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((2(3sin(2\u03c0x + 1) + 1))/(1(3sin(2\u03c0x + 1) + 1) - 2)) - 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) = -2(1cos(1\u03c0(-5x^2 + 3x - 1) - 3) - 1)^2 + 4(1cos(1\u03c0(-5x^2 + 3x - 1) - 3) - 1) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0(-5(-5x - 5) - 4) - 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-3sin(3\u03c0(-2x^2 - 2))) + 4 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) = -3sin(3\u03c0((2(2|x + 1| - 2) - 3)/(2(2|x + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(4(-3sin(1\u03c0x) - 3) + 5) - 5 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(-3sin(3\u03c0(1|x - 2| + 1) + 3) + 2)^2 - 1(-3sin(3\u03c0(1|x - 2| + 1) + 3) + 2) - 2 positive?\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) = -2cos(2\u03c0(-3|(5x - 4) - 1| + 2)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(-5(-4x^2 + 3x - 1) - 3) + 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1(-3cos(2\u03c0x + 3) - 3)^2 - 2(-3cos(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(-4(2|x - 1| + 2) + 5) - 1) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "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 + 3| - 2)^2 - 2(1|x + 3| - 2) + 5) - 1) - 2 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) = 5(1(-3sin(1\u03c0x + 3)) - 1)^2 - 1 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((2(2sin(3\u03c0x + 1) - 1) - 1)/(-1(2sin(3\u03c0x + 1) - 1) - 2))^2 - 4((2(2sin(3\u03c0x + 1) - 1) - 1)/(-1(2sin(3\u03c0x + 1) - 1) - 2)) - 4 is strictly increasing.\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) = -3sin(1\u03c0(3(-4x)) - 1) - 2 positive?\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) = 3sin(2\u03c0((-2((2x + 3)/(1x - 2)) + 2)/(-2((2x + 3)/(1x - 2)) - 2)) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-4(2x^2 - 2x - 2)^2 + 1(2x^2 - 2x - 2) + 3) + 3|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(1|((2x)/(-2x + 3)) - 2| - 1) + 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(5(-1cos(2\u03c0x + 1))^2 + 4(-1cos(2\u03c0x + 1)) - 2)^2 + 3(5(-1cos(2\u03c0x + 1))^2 + 4(-1cos(2\u03c0x + 1)) - 2) - 2 is strictly decreasing.\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) = -2|(-2cos(1\u03c0((-2x + 3)/(2x - 1)) - 2) + 2) - 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) = 1|((1(-1sin(1\u03c0x - 2) + 3) + 1)/(2(-1sin(1\u03c0x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2|(-1sin(2\u03c0x + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(2(-3cos(3\u03c0x + 1) + 1)^2 - 1(-3cos(3\u03c0x + 1) + 1) - 5) + 3)/(3(2(-3cos(3\u03c0x + 1) + 1)^2 - 1(-3cos(3\u03c0x + 1) + 1) - 5) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(2|(-3sin(1\u03c0x - 3) + 3) + 1| + 2) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3((-1(-3sin(1\u03c0x + 2) + 1) - 1)/(-2(-3sin(1\u03c0x + 2) + 1) + 3)) + 3)/(3((-1(-3sin(1\u03c0x + 2) + 1) - 1)/(-2(-3sin(1\u03c0x + 2) + 1) + 3)) - 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) = 5(-1sin(3\u03c0(-5x + 1) - 2) + 2)^2 + 5(-1sin(3\u03c0(-5x + 1) - 2) + 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3cos(2\u03c0(-1x^2 - 3x + 1) + 3) - 1)^2 - 1(3cos(2\u03c0(-1x^2 - 3x + 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": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-1cos(3\u03c0(2x^2 + 3x - 4) - 3) - 1) + 4 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) = -2sin(1\u03c0(-2|(1x - 4) + 2| + 1) - 3) + 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) = 1sin(2\u03c0(-1|(3|x - 3| + 2) + 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) = -2|(-3(2cos(3\u03c0x - 1) + 2)^2 - 2(2cos(3\u03c0x - 1) + 2) - 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) = -1sin(1\u03c0(4(1|x - 2| - 3)^2 + 5(1|x - 2| - 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": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0(1(-4x^2 + 2x - 1)^2 + 1) + 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(3|(1cos(3\u03c0x) + 1) + 1| - 2)^2 - 3(3|(1cos(3\u03c0x) + 1) + 1| - 2) - 5 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) = -3sin(2\u03c0((1((-2x - 3)/(-1x - 2)) - 3)/(-1((-2x - 3)/(-1x - 2)) - 3)) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(2sin(3\u03c0(1x - 3) - 2) - 3) - 3)/(-2(2sin(3\u03c0(1x - 3) - 2) - 3) - 2) 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) = -3(-2cos(1\u03c0(-2x + 3) - 2) + 3) - 1 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) = 1sin(1\u03c0(-1(-1x + 3)^2 - 2(-1x + 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": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(3sin(2\u03c0(3x^2 + 1x - 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": "58", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0((1(3x^2 - 3x - 1))/(1(3x^2 - 3x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2\u03c0(5((-3x)/(3x + 1)) - 2) - 3) - 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) = -3sin(2\u03c0(4(-4x^2 - 3x) - 5) + 3) + 2 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) = -4(1cos(2\u03c0(-3|x + 2| + 1) + 3)) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(2(-3sin(1\u03c0x - 1) + 3)^2 - 4(-3sin(1\u03c0x - 1) + 3) - 4))/(-3(2(-3sin(1\u03c0x - 1) + 3)^2 - 4(-3sin(1\u03c0x - 1) + 3) - 4) - 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) = 3sin(3\u03c0(-3(5x^2 - 1x)^2 + 2(5x^2 - 1x) - 4) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1cos(1\u03c0(5x^2 + 1) - 1))^2 + 2(-1cos(1\u03c0(5x^2 + 1) - 1)) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-4(-2sin(3\u03c0x - 1) + 3) + 2) - 1| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "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 - 5x + 3) + 2) + 1) - 2 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|(1sin(1\u03c0(-1x + 4) - 2) - 1) - 2| - 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) = -2((-1(-3cos(1\u03c0x - 1) - 2))/(-3(-3cos(1\u03c0x - 1) - 2) - 2))^2 - 2((-1(-3cos(1\u03c0x - 1) - 2))/(-3(-3cos(1\u03c0x - 1) - 2) - 2)) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(-3|(-4x^2 + 5x + 4) + 3| + 1) - 2) - 2 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) = -3(-5(-3cos(3\u03c0x - 3) - 3)^2 + 5(-3cos(3\u03c0x - 3) - 3) - 5) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(3\u03c0(-1|(1x + 5) - 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) = -2(-1(-2sin(2\u03c0x - 1))^2 + 2(-2sin(2\u03c0x - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2((2(-3cos(2\u03c0x + 2) + 3) - 2)/(-1(-3cos(2\u03c0x + 2) + 3) - 2)) - 2)/(-2((2(-3cos(2\u03c0x + 2) + 3) - 2)/(-1(-3cos(2\u03c0x + 2) + 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(-1sin(2\u03c0(-2x - 3) - 1) + 1) + 2)/(1(-1sin(2\u03c0(-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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-2(-3sin(2\u03c0x + 2) + 1) + 5)^2 + 1(-2(-3sin(2\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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(3sin(3\u03c0(-1|x| + 3) - 1) - 3) - 2)/(3(3sin(3\u03c0(-1|x| + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0((1(-2|x - 2| + 1) - 2)/(-3(-2|x - 2| + 1) - 1)) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3|(1sin(2\u03c0x - 2)) + 1| + 3) - 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) = 1cos(1\u03c0(1|(2|x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(1(-4x - 4) + 1) - 1) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3sin(1\u03c0(-1x^2 - 1x + 5) - 1) - 3)^2 - 2(-3sin(1\u03c0(-1x^2 - 1x + 5) - 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": "59", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2|(-1cos(2\u03c0x + 1) - 3) + 3| + 1) - 3)/(-3(-2|(-1cos(2\u03c0x + 1) - 3) + 3| + 1)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-3cos(2\u03c0(1x - 4) + 1)) - 3)/(-3(-3cos(2\u03c0(1x - 4) + 1)) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1(2|x + 2| - 2)^2 - 4(2|x + 2| - 2) + 2)^2 - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0((1((-1x + 2)/(1x - 3)) - 1)/(1((-1x + 2)/(1x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1sin(1\u03c0(-1x^2 - 3x + 1)) + 3)^2 + 3 positive?\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(-1|(-2cos(1\u03c0x + 3) + 1) + 2| - 1)^2 + 2(-1|(-2cos(1\u03c0x + 3) + 1) + 2| - 1) + 5 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(2(3cos(3\u03c0x + 2)) - 4)^2 + 1(2(3cos(3\u03c0x + 2)) - 4) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1cos(3\u03c0((2x + 1)/(2x + 3)) + 3) - 2) - 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) = -1|((1(3sin(1\u03c0x - 1) - 3) - 3)/(1(3sin(1\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2cos(3\u03c0(-5x + 5) - 2) - 2)^2 + 5(2cos(3\u03c0(-5x + 5) - 2) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0((1(-5x - 5) - 3)/(1(-5x - 5) + 3)) - 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) = -3((-1(-3cos(1\u03c0x + 2) + 2))/(-1(-3cos(1\u03c0x + 2) + 2) + 2))^2 - 3((-1(-3cos(1\u03c0x + 2) + 2))/(-1(-3cos(1\u03c0x + 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": "For the function f(x) = (-1(-1cos(3\u03c0(-1x^2 - 3x))))/(-1(-1cos(3\u03c0(-1x^2 - 3x))) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-5(-1x^2 + 3x + 1)^2 - 5(-1x^2 + 3x + 1) + 5) - 3| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(3cos(2\u03c0(3x^2 + 2x + 3) - 1) + 2) + 2)/(-2(3cos(2\u03c0(3x^2 + 2x + 3) - 1) + 2) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(-3|((2x - 2)/(-1x + 2)) + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((-3(1cos(2\u03c0x - 2) - 2) + 3)/(2(1cos(2\u03c0x - 2) - 2) + 3)) + 1| + 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) = 2(1sin(1\u03c0(-1|x + 2| + 3) + 1) - 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-1sin(1\u03c0((-1x + 1)/(-1x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((2(-1sin(2\u03c0x - 1) + 3) - 2)/(2(-1sin(2\u03c0x - 1) + 3))) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-5(-5x^2 + 4x + 4)^2 + 1(-5x^2 + 4x + 4) + 4) + 3| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-5(-3sin(2\u03c0x + 2))^2 - 1(-3sin(2\u03c0x + 2))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1(-1sin(3\u03c0x - 3) + 2) + 5)^2 - 2(-1(-1sin(3\u03c0x - 3) + 2) + 5) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-4(-1cos(3\u03c0x + 2) + 1)^2 - 2) - 1)/(-1(-4(-1cos(3\u03c0x + 2) + 1)^2 - 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(1\u03c0((1((2x + 3)/(1x - 2)))/(1((2x + 3)/(1x - 2)) + 1)) + 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0((-2(3x + 5))/(-2(3x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3(2cos(2\u03c0x)) - 5) - 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(-3cos(2\u03c0((3x - 3)/(-2x)) - 2) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1|(3sin(2\u03c0x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-3(2sin(2\u03c0x + 3) - 3) - 3)^2 + 5(-3(2sin(2\u03c0x + 3) - 3) - 3) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3(3sin(1\u03c0x + 2) - 1) + 5) - 4 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) = -3(3cos(2\u03c0(2|x - 3| - 3) + 1) - 2)^2 + 4(3cos(2\u03c0(2|x - 3| - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3cos(2\u03c0(-1x - 4) - 3))^2 - 3(-3cos(2\u03c0(-1x - 4) - 3)) + 4 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(1sin(2\u03c0(3|x + 3| - 2) - 1) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(1(-2sin(3\u03c0x + 1) - 1)^2 - 4(-2sin(3\u03c0x + 1) - 1) - 2) - 3| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-5(-2sin(1\u03c0x + 1) - 2)^2 + 2(-2sin(1\u03c0x + 1) - 2) + 4) + 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) = (-2(1cos(1\u03c0(-1x^2 + 1x + 1) - 3) - 2) + 1)/(-1(1cos(1\u03c0(-1x^2 + 1x + 1) - 3) - 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(4(-2sin(2\u03c0x - 3) + 2)^2 - 1(-2sin(2\u03c0x - 3) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(-2(-5x + 5)) - 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(2|(2sin(2\u03c0x) + 1)| + 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0((-1((2x + 2)/(3x - 2)) - 3)/(2((2x + 2)/(3x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-2cos(3\u03c0(-1x^2 + 4x) + 2) + 1) + 1)/(-1(-2cos(3\u03c0(-1x^2 + 4x) + 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": "36", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4((-1(3sin(3\u03c0x + 3)) - 1)/(-2(3sin(3\u03c0x + 3)) - 1))^2 - 4((-1(3sin(3\u03c0x + 3)) - 1)/(-2(3sin(3\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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-2cos(1\u03c0(3|x - 1| + 1) - 3) + 3)^2 + 4(-2cos(1\u03c0(3|x - 1| + 1) - 3) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(1(3x^2 + 2)^2 - 3(3x^2 + 2) + 1) + 1) + 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) = 2sin(2\u03c0(-2(5x^2 - 3x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1|(-1sin(2\u03c0x)) - 2|)^2 + 1(-1|(-1sin(2\u03c0x)) - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2(-1cos(3\u03c0x + 1) - 2)^2 - 1(-1cos(3\u03c0x + 1) - 2) - 1) + 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) = 1|((1(3sin(2\u03c0x - 3) - 3) - 3)/(2(3sin(2\u03c0x - 3) - 3) - 1)) + 2| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-4(-1cos(1\u03c0x + 1) + 1)^2 - 1(-1cos(1\u03c0x + 1) + 1) + 2) - 2| + 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) = 5(-1|(-2|x + 3| - 2) + 3| + 1)^2 - 2(-1|(-2|x + 3| - 2) + 3| + 1) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(3|(-1sin(2\u03c0x + 3) + 1) - 3| - 3)^2 + 2(3|(-1sin(2\u03c0x + 3) + 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|(3sin(3\u03c0(-1x + 3) - 3) + 1) - 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) = -1|(2|(-1sin(3\u03c0x + 1) - 1) - 3| - 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) = 2cos(1\u03c0((-3((-3x - 2)/(1x)) - 1)/(-1((-3x - 2)/(1x)) - 2)) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4((2(1cos(2\u03c0x + 3)) + 1)/(-1(1cos(2\u03c0x + 3)) - 3))^2 - 5((2(1cos(2\u03c0x + 3)) + 1)/(-1(1cos(2\u03c0x + 3)) - 3)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1(1sin(3\u03c0x + 1))^2 + 1) + 1)/(1(1(1sin(3\u03c0x + 1))^2 + 1) + 3) 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) = -2(-3cos(1\u03c0((-1x - 2)/(3x + 1)) - 3) + 3)^2 + 3(-3cos(1\u03c0((-1x - 2)/(3x + 1)) - 3) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1cos(3\u03c0(-1|x + 2| - 1) - 2) + 1)^2 + 1(-1cos(3\u03c0(-1|x + 2| - 1) - 2) + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1|(-1sin(3\u03c0x + 3) + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(-5(3|x + 2| + 3)^2 - 4(3|x + 2| + 3)) + 2) + 2 positive?\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(1sin(2\u03c0(-1|x + 1| + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1cos(2\u03c0(4x^2 + 1) + 2))^2 + 2(1cos(2\u03c0(4x^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": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-2cos(2\u03c0(-1x^2 - 4x) + 1)) - 4 is strictly increasing.\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) = -1sin(1\u03c0((-1(2|x + 1|) + 3)/(-1(2|x + 1|) + 2)) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3\u03c0((-3(4x + 1) - 3)/(-3(4x + 1))) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0((2(2x^2 - 2x) + 3)/(-1(2x^2 - 2x) + 2))) - 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) = 3(2cos(1\u03c0(1x + 4) + 2) - 1)^2 - 5(2cos(1\u03c0(1x + 4) + 2) - 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) = 3(2cos(2\u03c0(3|x + 1| + 3) + 3) - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(3|((-2x + 2)/(1x + 2)) - 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": "28", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0(-4((-2x + 3)/(3x - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-3(3sin(3\u03c0x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1cos(3\u03c0(-4x^2 - 4x - 3) - 2))^2 - 2(-1cos(3\u03c0(-4x^2 - 4x - 3) - 2)) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(2|(3cos(2\u03c0x - 1) + 1) + 1| - 3) - 1| - 3 negative?\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|(2sin(1\u03c0(-1x^2 + 4x + 2) - 3) + 1) - 2| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2sin(2\u03c0(4x + 1) - 1) - 2) 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(-1sin(2\u03c0x + 3) - 3)^2 - 1(-1sin(2\u03c0x + 3) - 3))| + 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) = -2|(-2(4x^2 + 4x - 5)^2 + 4)|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0(2(1x + 3)^2 - 5(1x + 3) + 4) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(1(-3cos(3\u03c0x) + 3) + 2)^2 + 5(1(-3cos(3\u03c0x) + 3) + 2) + 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) = -3(1sin(3\u03c0(-1x - 3)) - 2)^2 + 5(1sin(3\u03c0(-1x - 3)) - 2) - 5 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) = 3cos(1\u03c0(-2((-2x - 2)/(2x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1sin(2\u03c0(-4x^2 + 4x)) - 2)^2 + 4(1sin(2\u03c0(-4x^2 + 4x)) - 2) + 5 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(-2(-3x^2 + 3x + 2)^2 - 4(-3x^2 + 3x + 2) - 1)^2 - 1(-2(-3x^2 + 3x + 2)^2 - 4(-3x^2 + 3x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\u03c0(5((2x - 2)/(2x + 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": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0(-1(2x - 4) + 5) + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-2(-1cos(1\u03c0x - 3)) + 4)| - 3 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((3(3sin(2\u03c0x - 1) + 2) - 3)/(3(3sin(2\u03c0x - 1) + 2) - 1)) - 1)/(1((3(3sin(2\u03c0x - 1) + 2) - 3)/(3(3sin(2\u03c0x - 1) + 2) - 1)) + 3) 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|(-1|(2|x + 1| - 2)| + 1)| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(1sin(1\u03c0x - 3) - 2) + 3| + 1)^2 - 1(-1|(1sin(1\u03c0x - 3) - 2) + 3| + 1) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1cos(1\u03c0(2|x - 3| + 2) + 3) + 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2cos(1\u03c0(-2x^2 + 3x - 4) - 3)) + 1| - 1 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) = 1sin(1\u03c0(-2|((1x - 3)/(3x - 2)) + 1| + 3) + 2) + 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|(2sin(1\u03c0(-5x + 2) - 1) - 1) - 1| + 3 negative?\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) = (-3(-2cos(3\u03c0(-3x) + 3) + 3))/(3(-2cos(3\u03c0(-3x) + 3) + 3) + 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) = 4((-2(-3sin(2\u03c0x + 2) + 2) + 2)/(2(-3sin(2\u03c0x + 2) + 2) - 1))^2 - 1((-2(-3sin(2\u03c0x + 2) + 2) + 2)/(2(-3sin(2\u03c0x + 2) + 2) - 1)) - 4 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(3|(-2|x + 3| + 3) + 3|)^2 - 4(3|(-2|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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1(1sin(2\u03c0x + 2) + 2) + 4) + 3| - 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(-2sin(3\u03c0(4x - 3) + 1) - 2)^2 - 1(-2sin(3\u03c0(4x - 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": "22", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(3cos(3\u03c0((2x + 1)/(1x)) + 1) + 3) + 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) = (1(3sin(1\u03c0(2x - 3) + 2) + 2) - 3)/(2(3sin(1\u03c0(2x - 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) = -4(-1|(1cos(1\u03c0x + 3) + 1) - 3|)^2 - 4(-1|(1cos(1\u03c0x + 3) + 1) - 3|) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(2\u03c0(-1(-3x - 2) - 5) + 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) = 1cos(3\u03c0(-5(3x + 4)^2 - 1(3x + 4) - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-3|(-3cos(3\u03c0x + 1) + 3) - 3| - 2) + 1)/(-2(-3|(-3cos(3\u03c0x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-3sin(1\u03c0(-2|x + 1| - 1) - 3) + 3)^2 + 3(-3sin(1\u03c0(-2|x + 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": "27", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(3cos(1\u03c0(3x^2 + 4x - 4) + 3) + 1) + 3)/(1(3cos(1\u03c0(3x^2 + 4x - 4) + 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": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4((-2(-1cos(2\u03c0x - 3) - 3) - 1)/(-3(-1cos(2\u03c0x - 3) - 3) - 1)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-3sin(3\u03c0(-1x^2 - 5x) + 2) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-5(-3cos(3\u03c0x - 1)) + 4) - 1)/(3(-5(-3cos(3\u03c0x - 1)) + 4) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2(1cos(1\u03c0x - 3) - 3)^2 - 4(1cos(1\u03c0x - 3) - 3) - 1)^2 - 1(-2(1cos(1\u03c0x - 3) - 3)^2 - 4(1cos(1\u03c0x - 3) - 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) = -2|(1|(1cos(2\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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-5(-2cos(1\u03c0x + 2) + 3) - 4) + 1)/(1(-5(-2cos(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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(1\u03c0(3|((-2x - 3)/(-3x))| - 3)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2|(1cos(1\u03c0x - 2) - 2) - 2| + 1) + 5 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) = 2cos(2\u03c0(4((3x - 3)/(-1x + 2))^2 + 4((3x - 3)/(-1x + 2)) - 3) - 2) + 2 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) = -3|(-1|(1x^2 - 5x) - 1| + 2)| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1cos(1\u03c0(4x^2 + 5x - 5) + 1) - 2)^2 - 5(-1cos(1\u03c0(4x^2 + 5x - 5) + 1) - 2) - 5 is strictly increasing.\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) = 4(-2sin(2\u03c0(-4x) - 3) + 3) + 2 is strictly increasing.\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) = -2cos(3\u03c0(-4(3x^2 + 1x - 1) + 2) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1(3sin(1\u03c0x + 2) + 2)^2 + 4(3sin(1\u03c0x + 2) + 2) - 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2|(-1cos(1\u03c0x + 3) - 2) + 2| + 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((-3(-3sin(3\u03c0x + 2) + 1) - 1)/(1(-3sin(3\u03c0x + 2) + 1)))^2 - 4((-3(-3sin(3\u03c0x + 2) + 1) - 1)/(1(-3sin(3\u03c0x + 2) + 1))) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3((-1(-1sin(2\u03c0x - 2) - 3) + 2)/(2(-1sin(2\u03c0x - 2) - 3) + 2)) - 2)/(-2((-1(-1sin(2\u03c0x - 2) - 3) + 2)/(2(-1sin(2\u03c0x - 2) - 3) + 2)) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3cos(1\u03c0(1x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(1\u03c0(3(2|x - 2| + 1) - 2) + 2) 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) = -1sin(1\u03c0(-2|(-5x + 2) + 2| - 2) - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1sin(2\u03c0((-1x + 3)/(-3x + 2)) + 1) + 1)^2 - 4(1sin(2\u03c0((-1x + 3)/(-3x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1|(2cos(2\u03c0x + 2) + 2) - 1| - 2) + 1| + 2 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(3sin(2\u03c0x))^2 + 2(3sin(2\u03c0x)) - 1) - 3| + 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) = (2(-1(-3cos(1\u03c0x + 1) - 2)^2 + 2(-3cos(1\u03c0x + 1) - 2) + 1) + 2)/(-3(-1(-3cos(1\u03c0x + 1) - 2)^2 + 2(-3cos(1\u03c0x + 1) - 2) + 1) - 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) = 2cos(3\u03c0((-2(1x^2 - 5x - 2) + 1)/(2(1x^2 - 5x - 2))) + 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-1sin(2\u03c0(5x - 5) + 1) - 1) + 1)/(1(-1sin(2\u03c0(5x - 5) + 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": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(5(5x^2 - 1)^2 - 2(5x^2 - 1) - 1)| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(2cos(3\u03c0(-2|x + 2| - 2) + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(2sin(2\u03c0(-1|x - 2| + 2) + 3) - 3) + 1)/(-1(2sin(2\u03c0(-1|x - 2| + 2) + 3) - 3) - 1) 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) = -3sin(3\u03c0(1(2|x + 3| - 1)^2 + 4(2|x + 3| - 1)) - 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2sin(1\u03c0(3|x + 3| + 1) + 1) - 2) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1(3|x + 3| - 1)^2 + 4(3|x + 3| - 1) + 3)| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(1sin(3\u03c0(-3x^2 - 3x + 5) + 2) + 2) - 3)/(-2(1sin(3\u03c0(-3x^2 - 3x + 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": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(1(-2sin(3\u03c0x - 1) - 2)^2 + 3(-2sin(3\u03c0x - 1) - 2) - 5)^2 - 3(1(-2sin(3\u03c0x - 1) - 2)^2 + 3(-2sin(3\u03c0x - 1) - 2) - 5) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(-2|(2x^2 - 4x) + 3|) - 3) negative?\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) = 2(-1sin(3\u03c0(4x + 2) + 3) + 1)^2 - 1(-1sin(3\u03c0(4x + 2) + 3) + 1) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(1|(1cos(3\u03c0x) + 1) + 3| - 1) + 1)/(1(1|(1cos(3\u03c0x) + 1) + 3| - 1) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3\u03c0(2|(1x^2 - 4x + 3) + 2| - 3)) + 2 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) = 3cos(1\u03c0(5(2|x - 1| - 1) + 2) - 3) - 1 negative?\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) = 2cos(2\u03c0(2|(1x^2 - 5x - 3) + 2|) - 2) - 2 positive?\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) = (-3(3cos(1\u03c0(1|x| + 1))) - 1)/(2(3cos(1\u03c0(1|x| + 1))) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(2\u03c0(3(-3x^2 + 5x + 3) + 1) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2cos(3\u03c0(2|x + 3|) + 1) - 2)^2 - 1(-2cos(3\u03c0(2|x + 3|) + 1) - 2) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(4(-2sin(2\u03c0x + 1) - 1)^2 + 2(-2sin(2\u03c0x + 1) - 1) - 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3((2(3sin(2\u03c0x - 3) - 1) - 2)/(3(3sin(2\u03c0x - 3) - 1) - 1)) + 3)/(2((2(3sin(2\u03c0x - 3) - 1) - 2)/(3(3sin(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0(3(1x + 2)^2 + 3(1x + 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": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0(-3(4x - 4)^2 - 1(4x - 4) + 3) - 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-3|(-1x^2 - 5x - 2) - 3| + 3)^2 - 3(-3|(-1x^2 - 5x - 2) - 3| + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3sin(3\u03c0(-3x^2 - 1x - 1) + 2) - 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-5(1cos(3\u03c0x - 3) + 2) - 1) + 4 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) = -3((3(-2cos(1\u03c0x - 2) + 1) + 2)/(1(-2cos(1\u03c0x - 2) + 1)))^2 + 2((3(-2cos(1\u03c0x - 2) + 1) + 2)/(1(-2cos(1\u03c0x - 2) + 1))) + 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) = 3cos(1\u03c0((3(-3x - 4) - 3)/(-2(-3x - 4) - 3)) + 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(1|(-1x^2 + 4x - 1) + 3| - 2) + 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-1sin(1\u03c0(-2x)) - 2) + 3)/(2(-1sin(1\u03c0(-2x)) - 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) = 1cos(1\u03c0(1|((3x + 2)/(-2x + 3)) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(-3|(-2x^2 - 2x - 3) - 3| - 2) + 1) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-2sin(3\u03c0(1x - 5) + 1)) - 3| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(4(1sin(3\u03c0x - 3) + 3)^2 + 2(1sin(3\u03c0x - 3) + 3) - 3) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(3sin(2\u03c0(1x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(2|(-1x^2 + 5x - 1) - 2| - 1) + 3) + 1 is strictly decreasing.\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(5(-2sin(2\u03c0x + 1) - 1) + 4)^2 - 1(5(-2sin(2\u03c0x + 1) - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1(2sin(3\u03c0x - 1)) - 4)^2 - 5(1(2sin(3\u03c0x - 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) = 2cos(1\u03c0(4(1x^2 + 5x + 2)^2 + 3(1x^2 + 5x + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-1sin(2\u03c0(-5x^2 + 4x + 2) - 2) + 2) - 1)/(3(-1sin(2\u03c0(-5x^2 + 4x + 2) - 2) + 2)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(1(1sin(3\u03c0x - 3) + 1)^2 + 5(1sin(3\u03c0x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-1cos(3\u03c0(4x^2 + 3x - 2) - 1) + 1) - 2| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-5(2cos(3\u03c0x - 1) - 3) - 2) - 1)/(3(-5(2cos(3\u03c0x - 1) - 3) - 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) = 3cos(3\u03c0(-2(4x^2 - 2x - 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": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(2cos(1\u03c0(-2|x + 3| + 1)) + 3) + 3)/(-1(2cos(1\u03c0(-2|x + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2(1x^2 + 4x - 2)^2 - 3(1x^2 + 4x - 2) + 2)^2 - 2(-2(1x^2 + 4x - 2)^2 - 3(1x^2 + 4x - 2) + 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2sin(3\u03c0(1x^2 + 1x + 1) + 1)) + 2)/(3(2sin(3\u03c0(1x^2 + 1x + 1) + 1)) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1\u03c0(-2((-1x + 1)/(3x))^2 + 5((-1x + 1)/(3x)) + 2) + 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-4(-1cos(3\u03c0x - 2) + 1) + 1))/(-3(-4(-1cos(3\u03c0x - 2) + 1) + 1) + 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) = 2|(2sin(3\u03c0(-1x^2 + 3x + 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": "49", "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 + 5x + 5) + 2)/(1(-1x^2 + 5x + 5) - 3)) - 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|(3(-2|x - 1| + 1)^2 + 3(-2|x - 1| + 1) - 2) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-1sin(2\u03c0x - 1))^2 + 4(-1sin(2\u03c0x - 1)) - 4)^2 - 4(1(-1sin(2\u03c0x - 1))^2 + 4(-1sin(2\u03c0x - 1)) - 4) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3sin(3\u03c0((3x - 2)/(2x + 1))) + 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(1cos(3\u03c0(1|x + 1| + 1) - 3) - 2) + 3| - 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|(-1sin(2\u03c0((2x - 1)/(1x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(2sin(1\u03c0(1x^2 - 3x - 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": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0((2(-1x) + 1)/(1(-1x) - 2)) + 2) + 1 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) = 1|(-3cos(2\u03c0(4x - 1) - 2) + 2) - 2| - 3 is strictly decreasing.\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) = 1(2cos(1\u03c0((-1x + 3)/(2x)) - 3))^2 - 3(2cos(1\u03c0((-1x + 3)/(2x)) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2sin(1\u03c0(-1|x - 3| + 2) + 1) - 1)^2 + 4(-2sin(1\u03c0(-1|x - 3| + 2) + 1) - 1) - 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) = -2|(1sin(2\u03c0((-3x + 1)/(-3x - 2)) - 1) + 2) - 2| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3\u03c0(-2(3x^2 - 2x - 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) = (2(-3sin(2\u03c0((-1x - 3)/(2x + 2)) + 3) - 1) - 3)/(-1(-3sin(2\u03c0((-1x - 3)/(2x + 2)) + 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) = -4(4(-2sin(3\u03c0x) + 3)^2 + 2(-2sin(3\u03c0x) + 3) - 5) - 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) = 4(-4(-3cos(3\u03c0x + 3) - 2))^2 + 4(-4(-3cos(3\u03c0x + 3) - 2)) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(-5((-1x + 1)/(-3x)) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(3sin(3\u03c0(3x + 1) - 2)) - 3| - 1 negative?\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(-2|(5x - 1) + 1| + 1) - 1) + 1 positive?\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(3cos(2\u03c0(-1x^2 + 3x - 2) - 1) + 2) - 3)/(-3(3cos(2\u03c0(-1x^2 + 3x - 2) - 1) + 2) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0(1((-1x + 1)/(-1x)) + 2)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(-4(-4x - 4) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-1(1sin(3\u03c0x) - 2)^2 + 4) - 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2|(-2cos(3\u03c0x + 3) - 2) - 3| + 3)^2 - 5(2|(-2cos(3\u03c0x + 3) - 2) - 3| + 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) = 1cos(3\u03c0(2(-2x + 2) - 3) - 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(1(-2x^2 + 3x)^2 + 4(-2x^2 + 3x) - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(1\u03c0(-1((-2x - 2)/(3x + 2))^2 - 5((-2x - 2)/(3x + 2))) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1cos(2\u03c0((3x - 3)/(2x + 3)) + 3) + 3) + 3| - 3 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|(-1cos(2\u03c0(1x + 5) + 2)) - 1| - 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|(3(1sin(2\u03c0x - 3) - 1) - 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) = -2(1cos(1\u03c0(2x - 3) + 1) - 3) - 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(2(-1cos(2\u03c0x + 3) + 3) + 5) - 1)/(-1(2(-1cos(2\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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2cos(2\u03c0(3|x + 2| + 3) - 2) + 2) - 1)/(-2(-2cos(2\u03c0(3|x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(3\u03c0(1(-5x - 3)^2 - 1(-5x - 3) - 5) + 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) = -4(2|(-1cos(2\u03c0x + 1) + 3) + 1| + 1)^2 + 5(2|(-1cos(2\u03c0x + 1) + 3) + 1| + 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(4(1cos(3\u03c0x - 2) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2sin(3\u03c0((-2x + 2)/(3x + 2)) - 2) - 1) + 3 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) = 1sin(2\u03c0((1(-2x - 2) - 3)/(-2(-2x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(1(-1sin(3\u03c0x) + 2) + 5) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-4(-2x) + 5) + 1) + 1 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(1(2cos(1\u03c0x - 2) - 2)^2 - 5(2cos(1\u03c0x - 2) - 2) + 3) + 5 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|((3(2sin(3\u03c0x - 2) - 2))/(2(2sin(3\u03c0x - 2) - 2) + 1)) - 2| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3sin(3\u03c0(-5x + 1) + 3) + 1)^2 - 4(-3sin(3\u03c0(-5x + 1) + 3) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2\u03c0(1(-5x - 3)^2 - 4(-5x - 3) - 4) - 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(3|((1x + 1)/(1x)) + 2| + 3)) - 2 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) = -1cos(1\u03c0(1(2x)^2 - 3(2x) + 1) + 2) - 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) = -2|(-1(1cos(3\u03c0x - 3) + 3)^2 - 5(1cos(3\u03c0x - 3) + 3) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-1sin(2\u03c0(-2x - 5) + 2) + 3) + 2)/(-1(-1sin(2\u03c0(-2x - 5) + 2) + 3) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-2(-3sin(2\u03c0x + 3) - 1)^2 + 2(-3sin(2\u03c0x + 3) - 1) + 2) - 2| + 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(-3cos(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": "For the function f(x) = -3sin(2\u03c0(-3(1x + 5) + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3\u03c0((-3(2x))/(1(2x) + 3))) + 2 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) = -1cos(2\u03c0(-2(-4x + 4)^2 - 5(-4x + 4) + 1) + 2) + 1 positive?\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) = 5(1sin(1\u03c0((-2x + 3)/(1x)) - 3) - 1)^2 + 1(1sin(1\u03c0((-2x + 3)/(1x)) - 3) - 1) + 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) = (1(3sin(2\u03c0(-3|x + 1| - 2) + 3) + 2))/(-1(3sin(2\u03c0(-3|x + 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": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0(-2(-1|x| - 1) + 3) + 2) - 3 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((-2(-1cos(2\u03c0x - 2) - 3) - 3)/(3(-1cos(2\u03c0x - 2) - 3) - 2)) + 2)/(-1((-2(-1cos(2\u03c0x - 2) - 3) - 3)/(3(-1cos(2\u03c0x - 2) - 3) - 2)) + 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) = -3|(-3sin(2\u03c0(4x - 1) + 3) + 3) - 3| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-2sin(2\u03c0(-3x^2 - 5x - 3) + 2) - 3)| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(5(5x^2 + 5x - 4)^2 - 1(5x^2 + 5x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(2|(-2x^2 + 3x) - 2|) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(3cos(1\u03c0(-5x - 4) + 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": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0(4(-1x + 1)^2 + 3(-1x + 1) + 3)) - 1 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) = (2((-1(2cos(3\u03c0x + 1) - 3) - 2)/(3(2cos(3\u03c0x + 1) - 3) - 2)) - 3)/(1((-1(2cos(3\u03c0x + 1) - 3) - 2)/(3(2cos(3\u03c0x + 1) - 3) - 2)) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(-1|((-2x - 2)/(-3x - 1)) + 2| - 1) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-3(-1sin(2\u03c0x) + 3) + 2)/(-2(-1sin(2\u03c0x) + 3) + 3))^2 - 5((-3(-1sin(2\u03c0x) + 3) + 2)/(-2(-1sin(2\u03c0x) + 3) + 3)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-3(-2sin(2\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": "40", "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 + 2|) + 3) + 2) + 4 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) = 5(1sin(3\u03c0((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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1cos(1\u03c0(2x + 2) + 2) + 1)^2 - 1(-1cos(1\u03c0(2x + 2) + 2) + 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2\u03c0(-3(-2x + 2) - 3) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3sin(3\u03c0(-1x^2 - 4x - 1) - 3)) - 2)/(1(-3sin(3\u03c0(-1x^2 - 4x - 1) - 3))) 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) = -5(-1|(-3sin(1\u03c0x + 3) - 3) - 3|) - 4 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) = -2cos(3\u03c0((1(2x^2 + 4x + 5) + 2)/(2(2x^2 + 4x + 5) - 3)) + 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|((3(3sin(2\u03c0x + 1) - 2) - 2)/(1(3sin(2\u03c0x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0(-3|((3x)/(-2x - 2)) - 2|) + 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(2\u03c0(3((-1x + 3)/(2x + 3)) + 4) + 2) - 1 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) = 1cos(1\u03c0(4(2x^2 + 2x + 4)^2 + 3(2x^2 + 2x + 4) + 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": "13", "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 - 4) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(1\u03c0(1|((-2x - 1)/(-2x + 3)) + 2| + 2) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-2sin(2\u03c0(5x^2 - 3x - 5) - 3) - 2))/(2(-2sin(2\u03c0(5x^2 - 3x - 5) - 3) - 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-1(-3|x - 2| - 1)^2 - 3(-3|x - 2| - 1) + 5) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-3cos(3\u03c0((2x + 2)/(1x)) - 3) - 1) - 1| + 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) = 1sin(1\u03c0(-2|(1x + 4) - 3| - 1) + 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1|(3|x - 3| + 1) - 2| + 3)^2 + 5(-1|(3|x - 3| + 1) - 2| + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(2\u03c0((-2(-1x^2 - 5x + 2) + 3)/(-1(-1x^2 - 5x + 2) - 1)) + 2) - 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) = -3cos(2\u03c0(2|(-2|x - 3| + 2) + 1|) + 1) - 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) = 3|(2sin(2\u03c0(-4x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(5(1x^2 - 1x - 5)^2 - 1(1x^2 - 1x - 5) - 5)^2 + 1(5(1x^2 - 1x - 5)^2 - 1(1x^2 - 1x - 5) - 5) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-5(-2sin(3\u03c0x - 3) - 3))^2 - 3(-5(-2sin(3\u03c0x - 3) - 3)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(4(3sin(1\u03c0x - 3) - 2)^2 + 2(3sin(1\u03c0x - 3) - 2) - 2) - 3)/(2(4(3sin(1\u03c0x - 3) - 2)^2 + 2(3sin(1\u03c0x - 3) - 2) - 2) - 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) = 3cos(1\u03c0(2|(-5x^2 + 3x + 5)| - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((-2(1sin(2\u03c0x + 3) - 3) - 2)/(-2(1sin(2\u03c0x + 3) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((-2(3cos(3\u03c0x - 3) - 2) - 2)/(-3(3cos(3\u03c0x - 3) - 2) + 4))^2 + 3((-2(3cos(3\u03c0x - 3) - 2) - 2)/(-3(3cos(3\u03c0x - 3) - 2) + 4)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(1|(-1sin(3\u03c0x - 2) - 1) - 3| - 2) + 1)/(-1(1|(-1sin(3\u03c0x - 2) - 1) - 3| - 2) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1cos(2\u03c0((3x + 1)/(3x)) + 3) + 1) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3sin(3\u03c0(3|x + 1|) - 2)) - 1| - 3 positive?\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) = 1cos(3\u03c0((-1(3|x - 2|) + 2)/(-2(3|x - 2|) - 2)) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1sin(1\u03c0((-3x + 3)/(-1x + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1(-3cos(3\u03c0x + 2) + 2)^2 - 2(-3cos(3\u03c0x + 2) + 2) + 3))/(-1(1(-3cos(3\u03c0x + 2) + 2)^2 - 2(-3cos(3\u03c0x + 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": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-3sin(1\u03c0((-3x)/(2x + 3)) + 3) - 2) + 2)/(-3(-3sin(1\u03c0((-3x)/(2x + 3)) + 3) - 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|(-2|(-2cos(3\u03c0x - 3) - 2) - 3| + 3) + 1| - 2 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) = -3((1(1cos(3\u03c0x - 1) - 1))/(-1(1cos(3\u03c0x - 1) - 1) - 1)) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2cos(1\u03c0(5x^2 + 3x + 2) + 2) - 2) + 1 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) = 2sin(2\u03c0(5((2x - 3)/(1x - 3))^2 - 1((2x - 3)/(1x - 3))) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3\u03c0((3((-1x - 1)/(3x + 1)) + 1)/(3((-1x - 1)/(3x + 1)) - 1)) - 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-4(-1cos(2\u03c0x))^2 + 2(-1cos(2\u03c0x)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1sin(1\u03c0(5x - 2)) - 2) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(4(-2cos(1\u03c0x + 2) + 3)^2 - 4(-2cos(1\u03c0x + 2) + 3) - 3)^2 + 4(4(-2cos(1\u03c0x + 2) + 3)^2 - 4(-2cos(1\u03c0x + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2sin(1\u03c0(5x^2 - 5x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(1cos(3\u03c0(3|x - 2| + 1)) - 3) + 4 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) = 1(-2(3sin(2\u03c0x + 2) - 2)^2 - 3(3sin(2\u03c0x + 2) - 2) - 1)^2 + 5 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) = (2(1|(-2cos(1\u03c0x + 1) + 3) - 1| - 1) - 2)/(-3(1|(-2cos(1\u03c0x + 1) + 3) - 1| - 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) = -1cos(3\u03c0(4(-1|x + 1|) + 5) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-2sin(1\u03c0(-3x^2 + 4x + 3) - 3) + 1)^2 + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2cos(2\u03c0(5x + 4) - 2) + 1) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(-5((2x + 3)/(-1x))^2 + 5((2x + 3)/(-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": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2sin(1\u03c0(-1x + 2) + 2) + 3)^2 + 2(-2sin(1\u03c0(-1x + 2) + 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) = 3(-3sin(2\u03c0(-2|x - 3| - 3) - 2) - 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) = -5(-3cos(2\u03c0((-2x + 3)/(-2x - 2)) + 2) + 2) - 5 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) = -3(2sin(3\u03c0(3x^2 - 2x - 3) + 3) - 1)^2 + 2(2sin(3\u03c0(3x^2 - 2x - 3) + 3) - 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(1cos(2\u03c0(-1x^2 - 4x + 4) + 3) - 3) + 5 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) = -1sin(1\u03c0(-3(-3|x + 3| + 3)^2 - 4(-3|x + 3| + 3) - 2)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1sin(1\u03c0(-1|x + 3| - 2)) - 1) - 4 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(1cos(3\u03c0x - 3) - 2) + 5) - 4 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|(-1sin(3\u03c0(3x - 5) - 1) - 2)| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2|(3cos(3\u03c0x - 1) - 2) + 2| + 1) + 1)/(2(2|(3cos(3\u03c0x - 1) - 2) + 2| + 1) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0(-2(1x - 2)^2 - 1(1x - 2) - 5) + 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1(-3sin(2\u03c0x - 1) - 1) - 4) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(3\u03c0(-4(-5x^2 + 2x) - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(4(-2cos(3\u03c0x) - 1) - 4) - 4 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|(3cos(2\u03c0(1x - 4)) - 3) - 1| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0(-5((2x - 2)/(-3x)) - 5) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((3(2sin(3\u03c0x) + 2) + 1)/(2(2sin(3\u03c0x) + 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2\u03c0(4(3|x - 3| - 2) - 5) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-2|(1|x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-3cos(1\u03c0(3|x + 2| - 3) + 1) - 1)^2 + 1(-3cos(1\u03c0(3|x + 2| - 3) + 1) - 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1|(-2cos(2\u03c0x - 2) + 3) - 2| - 2)^2 + 2(-1|(-2cos(2\u03c0x - 2) + 3) - 2| - 2) + 4 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) = -2cos(1\u03c0(-1(-3x^2 + 3) + 1) + 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) = -2(2cos(2\u03c0(1|x + 1| + 1) - 1) - 2)^2 + 4(2cos(2\u03c0(1|x + 1| + 1) - 1) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-1|(2|x - 1| - 1) + 3|) + 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-2sin(3\u03c0(1|x| - 1) + 1) - 1) + 2)/(-2(-2sin(3\u03c0(1|x| - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(1(-2|x + 1| - 2)^2 - 3(-2|x + 1| - 2) - 5) + 1) - 1 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) = 3cos(2\u03c0(-5(-2x^2 + 5x - 2) + 3) - 1) - 1 is strictly increasing.\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) = -3sin(3\u03c0(-3(1|x - 1| - 3)^2 + 5(1|x - 1| - 3) - 4) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1\u03c0(4(1x^2 + 4x - 2)^2) + 3) - 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) = 3(2sin(1\u03c0(-2|x + 1| + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(3|(4x + 2) - 2| - 1)) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(-1|(5x - 5) - 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": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0((-1(-1|x + 1|) + 2)/(2(-1|x + 1|) - 1))) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1(2x^2 - 4x - 3)^2 - 3) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-3|(2cos(2\u03c0x - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2sin(3\u03c0(-3|x - 3| - 2) - 2) + 1) - 3)/(-3(2sin(3\u03c0(-3|x - 3| - 2) - 2) + 1) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(1(2sin(3\u03c0x - 3) + 1)^2 + 1(2sin(3\u03c0x - 3) + 1) + 3)^2 + 3(1(2sin(3\u03c0x - 3) + 1)^2 + 1(2sin(3\u03c0x - 3) + 1) + 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-2|(1cos(1\u03c0x - 3) + 3) + 1| - 2)^2 - 3(-2|(1cos(1\u03c0x - 3) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(2(-3sin(2\u03c0x + 1) + 3) - 5) - 3)/(-3(2(-3sin(2\u03c0x + 1) + 3) - 5) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}