{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-1(-2cos(1\u03c0x + 2) - 3)^2 + 1(-2cos(1\u03c0x + 2) - 3) + 5) - 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) = 1sin(3\u03c0((3(5x^2 - 5) + 2)/(-1(5x^2 - 5) - 2)) - 3) - 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) = -5(2|(2cos(1\u03c0((3x - 2)/(-3x + 4)))) + 1| - 3)^2 + 5(2|(2cos(1\u03c0((3x - 2)/(-3x + 4)))) + 1| - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1sin(2\u03c0(2|(1x^2 - 1x + 4) + 1| - 2) + 2)) - 5 is strictly decreasing.\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) = 3sin(3\u03c0((-1(2(-4x - 5)^2 + 1(-4x - 5) + 3) + 1)/(-3(2(-4x - 5)^2 + 1(-4x - 5) + 3) - 2)) - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2(1cos(1\u03c0((-1x + 3)/(2x - 3)) - 3)) - 4)^2 - 5(-2(1cos(1\u03c0((-1x + 3)/(2x - 3)) - 3)) - 4) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-2|(-4(3cos(2\u03c0x - 2) - 1)^2 + 1(3cos(2\u03c0x - 2) - 1) + 3) - 2| + 3) - 1)/(-1(-2|(-4(3cos(2\u03c0x - 2) - 1)^2 + 1(3cos(2\u03c0x - 2) - 1) + 3) - 2| + 3) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((2(4x^2 - 4x) - 2)/(-2(4x^2 - 4x) - 1)) + 2| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-3((1(-1|x - 3| + 2) - 3)/(-2(-1|x - 3| + 2)))^2 - 3((1(-1|x - 3| + 2) - 3)/(-2(-1|x - 3| + 2))) - 4) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(5((-1x + 3)/(-3x + 2)) + 3)^2 - 2(5((-1x + 3)/(-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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-3cos(1\u03c0x + 1) + 3)^2 + 5(-3cos(1\u03c0x + 1) + 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0(1(-1x^2 - 4x - 1) + 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": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(1|((3(5x^2 + 5x - 4) - 3)/(-2(5x^2 + 5x - 4))) - 2| - 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-1|x + 1| + 2)^2 - 4(-1|x + 1| + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(1|((2(2x + 3) + 2)/(-1(2x + 3))) + 1| - 3)^2 - 1(1|((2(2x + 3) + 2)/(-1(2x + 3))) + 1| - 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(5((-1x + 3)/(-3x))^2 - 2((-1x + 3)/(-3x)) - 2) - 1) + 1 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(-1(-1e^(1(-3|x + 1| + 2)) + 2)^2 - 2(-1e^(1(-3|x + 1| + 2)) + 2) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3cos(2\u03c0(2|x - 1| + 2) - 1) + 2) - 3)/(2(-3cos(2\u03c0(2|x - 1| + 2) - 1) + 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-1(2((-1x + 1)/(1x))^2 + 1((-1x + 1)/(1x)) + 2) + 3)| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((-1(1cos(3\u03c0x + 3) - 2) - 1)/(2(1cos(3\u03c0x + 3) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(3|((2x + 1)/(1x)) + 3| + 1) - 2) 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) = -2cos(2\u03c0((-3(3(-3|x + 1| + 3)^2 - 1(-3|x + 1| + 3) + 2) - 1)/(1(3(-3|x + 1| + 3)^2 - 1(-3|x + 1| + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(2x^2 - 2x - 5) + 3| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(-1(2x^2 - 2x + 4) - 4) + 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1(2sin(2\u03c0((3x - 1)/(-1x + 3)) - 1)) + 3)^2 + 4(-1(2sin(2\u03c0((3x - 1)/(-1x + 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": "For the function f(x) = -1|(3((1x - 3)/(-2x))) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2cos(1\u03c0x - 2) - 2) + 3)/(3(2cos(1\u03c0x - 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) = 2(1|((-2(2x^2 - 4x + 2) + 2)/(-1(2x^2 - 4x + 2) + 2)) + 3| + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1e^(2(-2|((-1(2x - 1) - 1)/(-2(2x - 1) + 3)) + 2| + 3)) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((2(-5(1|x + 3| - 2) + 3) - 3)/(3(-5(1|x + 3| - 2) + 3) + 2))^2 - 5((2(-5(1|x + 3| - 2) + 3) - 3)/(3(-5(1|x + 3| - 2) + 3) + 2)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-4(3|x| - 1) + 1)^2 + 4(-4(3|x| - 1) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-3cos(2\u03c0x - 1)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-5(2sin(1\u03c0x - 1))^2 + 5(2sin(1\u03c0x - 1)) - 5))/(2(-5(2sin(1\u03c0x - 1))^2 + 5(2sin(1\u03c0x - 1)) - 5) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(2cos(2\u03c0(-2x^2 - 1x - 5) - 2) + 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(2|(3sin(3\u03c0(-4x^2 - 3x - 5) - 3) - 2) - 1| - 2) + 1)/(-1(2|(3sin(3\u03c0(-4x^2 - 3x - 5) - 3) - 2) - 1| - 2) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(1\u03c0(-1x - 4) + 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) = 3cos(1\u03c0(-3|x + 1| - 1) + 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) = (-1(-1(-5(1|x + 2| - 3) + 1)^2 + 2(-5(1|x + 2| - 3) + 1) - 2) - 3)/(-1(-1(-5(1|x + 2| - 3) + 1)^2 + 2(-5(1|x + 2| - 3) + 1) - 2) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(-1|x - 2| - 2) - 1) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(3sin(1\u03c0x + 3) - 1) - 3)/(2(3sin(1\u03c0x + 3) - 1) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|((-2(-2x - 1) + 3)/(2(-2x - 1) - 1)) - 3| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(-3|x - 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) = 1(2|(5((2x)/(2x + 2)) - 2)| - 3)^2 - 5(2|(5((2x)/(2x + 2)) - 2)| - 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2\u03c0(-3x - 3) + 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-1cos(1\u03c0(-2|(2x) + 2| - 2) - 3) + 2) + 2)/(-1(-1cos(1\u03c0(-2|(2x) + 2| - 2) - 3) + 2) - 2) 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) = -2|(2((1x + 2)/(3x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3\u03c0(4x^2 + 3x - 1) - 2) - 1 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) = -5(2sin(1\u03c0(-1|x| + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0(-1((1(-5x^2 - 5x + 4) - 3)/(-1(-5x^2 - 5x + 4) - 1)) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3\u03c0(-5(1|x + 3| + 1)^2 - 3(1|x + 3| + 1) - 3)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(3(2e^(-1(2|x - 2| - 1)) - 3))^2 - 3(3(2e^(-1(2|x - 2| - 1)) - 3)) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "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 + 3| - 2)^2 - 5(3|x + 3| - 2) + 1) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|((-3(3(3x - 1)^2 + 1(3x - 1) - 4) - 3)/(1(3(3x - 1)^2 + 1(3x - 1) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1cos(2\u03c0x + 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) = 2(-3|x - 1| + 1)^2 + 5(-3|x - 1| + 1) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(1(3|(-2e^(1x) + 3) - 1| - 2)^2 - 2(3|(-2e^(1x) + 3) - 1| - 2) + 4) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1((1(3sin(2\u03c0x - 3) - 3) + 3)/(1(3sin(2\u03c0x - 3) - 3)))^2 + 3((1(3sin(2\u03c0x - 3) - 3) + 3)/(1(3sin(2\u03c0x - 3) - 3))) - 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(1\u03c0(1(1(-2|x - 3| - 3)^2 - 2(-2|x - 3| - 3) + 4) + 4) + 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2cos(1\u03c0x - 1) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((3x + 2)/(-2x + 3))^2 + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2((-2(3x^2 - 1))/(2(3x^2 - 1) - 3)) - 5) + 1| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(3((1x - 2)/(-3x - 1))^2 + 5((1x - 2)/(-3x - 1)) + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-5((3(4x + 2) + 1)/(-1(4x + 2) - 2))^2 - 5((3(4x + 2) + 1)/(-1(4x + 2) - 2)) + 3) - 2| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1|(-1cos(2\u03c0x - 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": "For the function f(x) = 3(-3(-3cos(3\u03c0x + 1) + 1) + 1)^2 + 2(-3(-3cos(3\u03c0x + 1) + 1) + 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0(-2|x| + 3) + 2) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-3e^(1(-2((2x + 2)/(-3x + 2)) + 3)) + 1)^2 + 2(-3e^(1(-2((2x + 2)/(-3x + 2)) + 3)) + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(5((-1x)/(-3x - 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": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0(-2x^2 + 1x - 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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1(-2|x + 1| + 3)^2 - 5(-2|x + 1| + 3) - 5) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(5(3sin(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0((-3(-5(-1x^2 + 4x - 5) - 5) - 1)/(3(-5(-1x^2 + 4x - 5) - 5) - 1)) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2(-2((1x + 3)/(-2x + 3)) + 1)^2 - 5)| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-5(-3x^2 - 4x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-3cos(3\u03c0x + 2))^2 + 5(-3cos(3\u03c0x + 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) = (3(2|(-1x^2 - 4x + 1) - 2| - 3))/(-1(2|(-1x^2 - 4x + 1) - 2| - 3) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(2|(3x^2 - 4) + 2| - 1) + 2)/(-1(2|(3x^2 - 4) + 2| - 1) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2cos(1\u03c0x + 2) - 1)^2 - 4(-2cos(1\u03c0x + 2) - 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((3(-1|x + 1| + 3) + 3)/(-1(-1|x + 1| + 3)))^2 + 1((3(-1|x + 1| + 3) + 3)/(-1(-1|x + 1| + 3))) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3sin(2\u03c0x + 1) - 2) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(1\u03c0(3((-2(-1x - 1) + 3)/(1(-1x - 1) + 1))^2 + 3((-2(-1x - 1) + 3)/(1(-1x - 1) + 1)) + 2) + 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(5x^2 + 4x) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-4(3cos(2\u03c0(2|x - 1| + 1) + 2) + 3)^2 - 1(3cos(2\u03c0(2|x - 1| + 1) + 2) + 3) - 3) + 3)/(2(-4(3cos(2\u03c0(2|x - 1| + 1) + 2) + 3)^2 - 1(3cos(2\u03c0(2|x - 1| + 1) + 2) + 3) - 3) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((1(-2cos(3\u03c0x)) + 2)/(3(-2cos(3\u03c0x)) - 1))^2 + 5((1(-2cos(3\u03c0x)) + 2)/(3(-2cos(3\u03c0x)) - 1)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-3sin(1\u03c0((3x + 1)/(2x - 3)) - 3) - 1)^2 + 5(-3sin(1\u03c0((3x + 1)/(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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1|(-1(1cos(3\u03c0x) + 1) + 2) + 2|)^2 + 3(1|(-1(1cos(3\u03c0x) + 1) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3e^(-1(-5(3x + 3)^2 + 2(3x + 3) - 1)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(5(-2|(-1x + 4) - 1| + 2)^2 + 2(-2|(-1x + 4) - 1| + 2) + 5) + 1)/(-1(5(-2|(-1x + 4) - 1| + 2)^2 + 2(-2|(-1x + 4) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1cos(2\u03c0(3|x|) - 3) + 3) - 2)/(2(1cos(2\u03c0(3|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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1(5((3x - 3)/(-2x - 2))^2 + 1((3x - 3)/(-2x - 2)) - 2) - 5) + 1| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3\u03c0(5((-2x + 1)/(-2x + 3))^2 - 4((-2x + 1)/(-2x + 3)) - 1) - 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-3(-3sin(1\u03c0x)) + 3)/(-1(-3sin(1\u03c0x)) + 3))^2 + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(2cos(2\u03c0x) - 1) - 3| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0((2(4x))/(-1(4x) + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(1(1(-1|x + 2|)^2 + 2(-1|x + 2|) + 1) + 2) - 3)/(-1(1(1(-1|x + 2|)^2 + 2(-1|x + 2|) + 1) + 2) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(1(5(-1|x + 2| + 1)^2 - 5(-1|x + 2| + 1) + 5) + 5) + 3)/(2(1(5(-1|x + 2| + 1)^2 - 5(-1|x + 2| + 1) + 5) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-1(-1cos(1\u03c0(-1|x - 1| + 1) + 3) - 2) - 4)^2 + 4(-1(-1cos(1\u03c0(-1|x - 1| + 1) + 3) - 2) - 4) - 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) = 2|(4x^2 + 3x + 2) - 3| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-5(1e^(2(-1x)) - 1)^2 - 4(1e^(2(-1x)) - 1) + 5)| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2cos(1\u03c0(3(-3|x + 1|)^2 + 1(-3|x + 1|) + 2) - 3) + 2))/(3(2cos(1\u03c0(3(-3|x + 1|)^2 + 1(-3|x + 1|) + 2) - 3) + 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) = 3|(1(-1((1x - 1)/(2x - 3)) + 3)^2 - 2(-1((1x - 1)/(2x - 3)) + 3) + 4) - 2| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(2((-3x + 2)/(2x - 1))^2 - 3((-3x + 2)/(2x - 1)) - 4) + 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((1(1|(2x + 3) - 1| + 1) + 1)/(2(1|(2x + 3) - 1| + 1) - 3))^2 - 3((1(1|(2x + 3) - 1| + 1) + 1)/(2(1|(2x + 3) - 1| + 1) - 3)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1|(5(-2cos(1\u03c0x + 1) + 3)^2 + 4(-2cos(1\u03c0x + 1) + 3) - 3) + 1| + 3) + 2)/(-2(1|(5(-2cos(1\u03c0x + 1) + 3)^2 + 4(-2cos(1\u03c0x + 1) + 3) - 3) + 1| + 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) = 1|(3x^2 - 5x - 5) - 1| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1(-4(1|x|)^2 + 1(1|x|) + 4) - 4) + 2)/(-3(-1(-4(1|x|)^2 + 1(1|x|) + 4) - 4) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1e^(-1(-4x^2 - 2x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1|x + 2| - 1)^2 + 4(-1|x + 2| - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3|((1(2x + 1) - 3)/(-1(2x + 1) + 2)) + 2| + 3)^2 + 1(3|((1(2x + 1) - 3)/(-1(2x + 1) + 2)) + 2| + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(-1x^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": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((3(-1cos(1\u03c0x - 1) - 2) + 3)/(-1(-1cos(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(1\u03c0((-1(1x + 1) - 3)/(2(1x + 1) + 1)) + 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(2|(5(3cos(1\u03c0x - 2) - 2)^2 + 3(3cos(1\u03c0x - 2) - 2) + 3) + 3| + 3))/(-3(2|(5(3cos(1\u03c0x - 2) - 2)^2 + 3(3cos(1\u03c0x - 2) - 2) + 3) + 3| + 3) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(1\u03c0(1x - 3) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1cos(2\u03c0((-1x)/(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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-3|(-1(1cos(2\u03c0x - 2) + 3)^2 + 4)| - 1))/(-3(-3|(-1(1cos(2\u03c0x - 2) + 3)^2 + 4)| - 1) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-3(1cos(2\u03c0((-3x)/(-3x + 1)) - 3) + 1)^2 + 2(1cos(2\u03c0((-3x)/(-3x + 1)) - 3) + 1) + 4) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((1(3cos(1\u03c0(2x^2 - 3x + 3) - 3) - 1) + 3)/(3(3cos(1\u03c0(2x^2 - 3x + 3) - 3) - 1) - 1)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-4((-1x - 1)/(3x + 1))^2 - 3((-1x - 1)/(3x + 1)))| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0((-3(4(5x^2 + 2x - 2) - 5) - 3)/(1(4(5x^2 + 2x - 2) - 5))) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-5x^2 - 4x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4((2(-2sin(2\u03c0x) - 2) + 3)/(-2(-2sin(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3\u03c0(2x^2 - 2) - 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(4((1x + 2)/(-3x - 2)) - 1)| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3\u03c0(-5x + 3) + 1) + 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) = -1|(3sin(1\u03c0x) - 2) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1e^(2(-5(4x^2 - 4x - 2) - 5)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0((1x - 2)/(-2x + 1)) - 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(3((-1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)) - 5) - 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(2|x + 3| - 3) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4((-3x + 1)/(-1x))^2 + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(3sin(3\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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(3|(-3x + 3)| - 1)^2 + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-2(1cos(1\u03c0(2|x - 1| + 3) + 3) + 1) + 5)^2 + 3(-2(1cos(1\u03c0(2|x - 1| + 3) + 3) + 1) + 5) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((-2(2|x - 3|) - 1)/(1(2|x - 3|) - 3))^2 - 3((-2(2|x - 3|) - 1)/(1(2|x - 3|) - 3)) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-3(-3cos(3\u03c0x - 3)) - 1)/(-1(-3cos(3\u03c0x - 3)) + 2)) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5((-3(4x + 4) - 1)/(3(4x + 4) + 3))^2 - 1((-3(4x + 4) - 1)/(3(4x + 4) + 3)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-1(2cos(1\u03c0x) - 1) - 1)^2 + 2(-1(2cos(1\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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-4(-3cos(3\u03c0x + 3) + 2)^2 - 2(-3cos(3\u03c0x + 3) + 2) + 2)| 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(-5(-2|x| + 1) + 3)^2 - 4(-5(-2|x| + 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((-2(-3x - 5) - 3)/(-3(-3x - 5) + 1))^2 - 2((-2(-3x - 5) - 3)/(-3(-3x - 5) + 1)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-1(5(-1x^2 + 4x - 3) - 4) - 1)/(-2(5(-1x^2 + 4x - 3) - 4) + 1)) - 2| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-2|(1x - 4) - 2| + 1)^2 + 1(-2|(1x - 4) - 2| + 1) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-2(1x + 3)^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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-5(-2(-2|x + 2| + 3))^2 - 5(-2(-2|x + 2| + 3))) - 1)/(1(-5(-2(-2|x + 2| + 3))^2 - 5(-2(-2|x + 2| + 3))) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(2(-1sin(3\u03c0(5x^2 + 5x - 2) + 1) - 3) - 3) + 3)/(-3(2(-1sin(3\u03c0(5x^2 + 5x - 2) + 1) - 3) - 3)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-2|(2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-3(-4(-1x - 4)^2 + 1(-1x - 4) - 4))/(2(-4(-1x - 4)^2 + 1(-1x - 4) - 4) + 2)) + 2| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2|(-3((-1x + 3)/(-1x + 2)) + 2) - 1| + 2)^2 + 5(-2|(-3((-1x + 3)/(-1x + 2)) + 2) - 1| + 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0(2x + 4) - 2) - 3 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) = 1|(-1cos(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": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(-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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-3(1cos(1\u03c0x + 1) + 1) - 3)/(2(1cos(1\u03c0x + 1) + 1) + 1))^2 - 3((-3(1cos(1\u03c0x + 1) + 1) - 3)/(2(1cos(1\u03c0x + 1) + 1) + 1)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(5x + 2) + 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(3|(-2x^2 - 2x + 1) - 1| - 2) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3(-2|x - 1|)^2 + 1(-2|x - 1|) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(1sin(3\u03c0x) + 1) + 1)/(-1(1sin(3\u03c0x) + 1) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(1e^(1(4x^2 - 5x - 5))))/(-1(1e^(1(4x^2 - 5x - 5))) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(1(1(-3|x + 2|) + 5)^2 + 3(1(-3|x + 2|) + 5) - 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": "54", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(1\u03c0(-5((3x)/(1x + 1)) + 4) - 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-3|((-1(-5x^2 + 2x + 1) - 3)/(-1(-5x^2 + 2x + 1) + 2)) + 2|) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2|(4((-2x - 2)/(-2x))^2 + 5((-2x - 2)/(-2x)) - 4) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((2(1x + 4))/(-2(1x + 4) + 3)) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(4(-4((3x - 2)/(2x + 2))^2 - 5) - 2) - 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(1|x - 3|)^2 - 3(1|x - 3|) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(2|x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((-1(1|x - 1| - 3) + 3)/(-2(1|x - 1| - 3) - 2))^2 - 4((-1(1|x - 1| - 3) + 3)/(-2(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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(3cos(3\u03c0x + 2) + 1)^2 - 3(3cos(3\u03c0x + 2) + 1) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(2sin(1\u03c0((2x + 1)/(2x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3\u03c0(3|x + 2| - 2) + 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3cos(2\u03c0(-1|x + 3| - 2) - 2) - 1) + 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) = 4(-2sin(1\u03c0x - 1) - 3)^2 + 3(-2sin(1\u03c0x - 1) - 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) = 1((-2(1|x + 1| - 1))/(3(1|x + 1| - 1) - 2))^2 - 2((-2(1|x + 1| - 1))/(3(1|x + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((3(1(3|x - 2| - 2)) + 2)/(-1(1(3|x - 2| - 2))))^2 + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(3sin(3\u03c0x)) + 3| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0((1(-2x^2 + 1x + 4) + 3)/(3(-2x^2 + 1x + 4) + 2)) + 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-2cos(3\u03c0(2x + 3) + 2) - 1))/(-1(-2cos(3\u03c0(2x + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(3(-2e^(-1(3|x + 2|)) + 1)^2 - 4(-2e^(-1(3|x + 2|)) + 1) - 4) - 1)/(-1(3(-2e^(-1(3|x + 2|)) + 1)^2 - 4(-2e^(-1(3|x + 2|)) + 1) - 4) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(1sin(2\u03c0(-3|x - 1|) + 3) + 2)^2 - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((3(1e^(-1(3x + 2)) - 2))/(-1(1e^(-1(3x + 2)) - 2) + 2))^2 + 4((3(1e^(-1(3x + 2)) - 2))/(-1(1e^(-1(3x + 2)) - 2) + 2)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((-2x)/(-3x - 1))^2 + 2((-2x)/(-3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(-1(2x + 3)^2 + 3(2x + 3)) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1cos(1\u03c0(-1((2x + 2)/(-3x)) - 5) + 2) - 3)^2 - 3(-1cos(1\u03c0(-1((2x + 2)/(-3x)) - 5) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-1sin(1\u03c0x + 2) - 2)^2 + 2(-1sin(1\u03c0x + 2) - 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-3|x + 1| + 2)^2 + 2(-3|x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(1x^2 + 2x) + 3) + 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(3cos(1\u03c0(4x + 4) + 3) - 1) - 3)/(-1(3cos(1\u03c0(4x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1sin(1\u03c0(3(3|x - 3| + 3) - 2) + 1) - 1)^2 - 1(-1sin(1\u03c0(3(3|x - 3| + 3) - 2) + 1) - 1) - 2 positive?\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) = (1(3(2|(-2x^2 + 3x + 3) + 2|) + 4) - 3)/(-3(3(2|(-2x^2 + 3x + 3) + 2|) + 4) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(-2x^2 - 1x - 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": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3e^(1(2|x + 3| - 2)) - 2)^2 - 3(3e^(1(2|x + 3| - 2)) - 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1|(-2cos(1\u03c0(1x - 3) - 1) + 1)| + 1)^2 + 4(-1|(-2cos(1\u03c0(1x - 3) - 1) + 1)| + 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(2(3|(-2x^2 - 3x - 3) + 2| - 2) - 5) - 1)/(-1(2(3|(-2x^2 - 3x - 3) + 2| - 2) - 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1|x - 1| + 3)^2 - 4(-1|x - 1| + 3) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-3(-2sin(3\u03c0(5x^2 - 5x - 4) - 1) - 1))| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2((-1(3cos(1\u03c0x - 2) + 2) + 1)/(-1(3cos(1\u03c0x - 2) + 2) - 2))^2 - 1((-1(3cos(1\u03c0x - 2) + 2) + 1)/(-1(3cos(1\u03c0x - 2) + 2) - 2)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1e^(1(-3(1|((2x - 1)/(2x + 2)) - 3| - 3)^2 - 3(1|((2x - 1)/(2x + 2)) - 3| - 3) + 4)) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0((2x + 2)/(-2x)) + 3) + 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) = 1(-3|x - 1| - 2)^2 + 5(-3|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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1((-3(-1sin(1\u03c0(1|x + 2| - 2) - 3) - 2) + 2)/(1(-1sin(1\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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2e^(2(3(-1|x|)^2 - 1(-1|x|) - 3)) + 2))/(-2(2e^(2(3(-1|x|)^2 - 1(-1|x|) - 3)) + 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-3|(-1(2cos(2\u03c0x - 3) - 1)^2 + 3(2cos(2\u03c0x - 3) - 1) - 4) - 2| - 1) + 2)/(-2(-3|(-1(2cos(2\u03c0x - 3) - 1)^2 + 3(2cos(2\u03c0x - 3) - 1) - 4) - 2| - 1)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((1(1(-2sin(1\u03c0x - 1) - 3)^2 + 2(-2sin(1\u03c0x - 1) - 3) - 1))/(2(1(-2sin(1\u03c0x - 1) - 3)^2 + 2(-2sin(1\u03c0x - 1) - 3) - 1) + 1)) - 1| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(3\u03c0((2x - 3)/(-3x - 2))) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2(1|x + 1| - 2))^2 + 1(2(1|x + 1| - 2)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(1sin(1\u03c0x + 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) = -2sin(1\u03c0(-2x^2 + 4x + 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|((-2(-2x^2 - 4x + 1))/(-2(-2x^2 - 4x + 1) - 3)) - 2| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(5(1x^2 - 4x - 4) + 2) - 1| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((2(-1|x - 1|) - 1)/(-1(-1|x - 1|) - 1))^2 - 5((2(-1|x - 1|) - 1)/(-1(-1|x - 1|) - 1)) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2|(1((2x)/(3x - 1))) + 2| + 3)^2 - 1(-2|(1((2x)/(3x - 1))) + 2| + 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0((3x - 1)/(-1x - 3))) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1\u03c0(1(1|((-2x)/(1x + 3)) + 3|) + 3) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1|((-3x - 1)/(1x + 1)) - 3|) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((3(-1(4x - 3)^2 - 1) - 1)/(3(-1(4x - 3)^2 - 1) + 3)) - 2| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((-2(-5(1x^2 - 4x - 4) + 4) + 2)/(1(-5(1x^2 - 4x - 4) + 4) - 3)) + 1| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3\u03c0((3(-3x - 3) + 3)/(2(-3x - 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) = 2sin(1\u03c0((2(5x^2 - 1x + 2) - 3)/(1(5x^2 - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1sin(3\u03c0((-2x + 1)/(3x - 1)) + 1) + 1) - 3| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(-1sin(3\u03c0x - 1) + 2)^2 + 5(-1sin(3\u03c0x - 1) + 2) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(1\u03c0(-1|(1x - 1) + 1|) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(1sin(1\u03c0x + 3) + 2) + 2)/(-3(1sin(1\u03c0x + 3) + 2)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-1(-4(-3|x - 3| + 1) + 2) + 3)/(-2(-4(-3|x - 3| + 1) + 2) + 1))^2 + 4((-1(-4(-3|x - 3| + 1) + 2) + 3)/(-2(-4(-3|x - 3| + 1) + 2) + 1)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0(-1|((3(-3x^2 + 3x + 4) + 3)/(-1(-3x^2 + 3x + 4) + 3)) - 2| + 3) + 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((-3(-3|(-5x^2 - 1x) + 3|) + 3)/(2(-3|(-5x^2 - 1x) + 3|) + 2)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2sin(1\u03c0x - 3) + 2) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-1sin(3\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": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(2|(-5((1x - 1)/(-2x)) - 1) + 3| + 1)^2 + 2(2|(-5((1x - 1)/(-2x)) - 1) + 3| + 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2((2(-2x + 1) - 2)/(2(-2x + 1) + 2))^2 + 2((2(-2x + 1) - 2)/(2(-2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1sin(3\u03c0x + 1) + 1) - 2)/(3(-1sin(3\u03c0x + 1) + 1) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(-4(-2|x - 1| + 1)) + 1) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((1(-2(-3cos(2\u03c0x - 3) - 3) + 3) - 1)/(-1(-2(-3cos(2\u03c0x - 3) - 3) + 3) + 2))^2 + 2((1(-2(-3cos(2\u03c0x - 3) - 3) + 3) - 1)/(-1(-2(-3cos(2\u03c0x - 3) - 3) + 3) + 2)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-3(-2(-1x^2 + 4x + 2) + 4) + 1)/(-3(-2(-1x^2 + 4x + 2) + 4) - 2)) - 3| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(3\u03c0(-1(1|((-2x)/(-2x - 2)) + 1| + 3)^2 - 1(1|((-2x)/(-2x - 2)) + 1| + 3) + 3) - 1) - 2 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) = (-2(-4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 4) - 2)/(-2(-4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 4) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(-1|x + 1| - 3) + 1) + 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) = 1sin(3\u03c0(-2(-5x - 5)^2 + 3(-5x - 5) + 2) + 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2cos(2\u03c0x + 3) - 3) + 4 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(-1|x + 1| + 2)^2 + 4(-1|x + 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": "2", "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) + 2) - 1| + 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(5x - 4) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2((1x - 1)/(-3x - 1))^2 + 4((1x - 1)/(-3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(1\u03c0(3x + 4) - 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1sin(1\u03c0(5(1x)^2 - 3) + 3) - 1) - 1| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(3cos(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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0(-2|(-5x^2 - 2x - 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": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2(-5x^2 + 1x + 5)) + 3| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-3|x - 3| + 3)^2 - 1(-3|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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-3sin(1\u03c0(4x + 3) + 1) + 2) - 1)/(-2(-3sin(1\u03c0(4x + 3) + 1) + 2) - 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) = 3cos(2\u03c0(1|(4(-5x^2 - 4x + 4) - 5) - 1| - 3) + 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) = 1|(-1(-2sin(3\u03c0x + 1) - 1) - 4) - 1| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(3sin(2\u03c0x + 1) - 2) + 1)/(-3(3sin(2\u03c0x + 1) - 2)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2((-1(-2|x + 2| + 3) - 2)/(-1(-2|x + 2| + 3) - 3))^2 - 1((-1(-2|x + 2| + 3) - 2)/(-1(-2|x + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-4(2cos(3\u03c0x - 3) - 1)^2 + 4(2cos(3\u03c0x - 3) - 1) - 5) + 3)/(2(-4(2cos(3\u03c0x - 3) - 1)^2 + 4(2cos(3\u03c0x - 3) - 1) - 5) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((-2x + 2)/(-2x - 1))^2 + 1((-2x + 2)/(-2x - 1)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-4(2|((-1x - 1)/(-2x + 3)) - 3| - 3) + 5)^2 - 1(-4(2|((-1x - 1)/(-2x + 3)) - 3| - 3) + 5) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((-3(2sin(1\u03c0x) + 3))/(3(2sin(1\u03c0x) + 3) + 3)) - 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) = 3|(3cos(3\u03c0(-3((2x + 3)/(2x + 1))^2 - 2((2x + 3)/(2x + 1)) + 2) + 1) - 1) - 3| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(1\u03c0(2|((-2(1x - 4))/(2(1x - 4) - 1)) - 3| + 2) + 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-2cos(1\u03c0(-2x) + 2)) - 2| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0(4x^2 - 2x - 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": "54", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(2|(-5x^2 - 4x + 5) + 3| + 1) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2(4x^2 - 5x + 1) - 1) + 2| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-1cos(1\u03c0((3x - 1)/(-1x + 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": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-1|(4(5x^2 - 2x) - 2) - 3| + 3) + 2)/(-1(-1|(4(5x^2 - 2x) - 2) - 3| + 3) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-3sin(1\u03c0(-5x + 3)) + 1) - 1)/(2(-3sin(1\u03c0(-5x + 3)) + 1) + 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) = (2(-1(1|(3x) - 3|)^2 + 3(1|(3x) - 3|) + 4) - 3)/(-1(-1(1|(3x) - 3|)^2 + 3(1|(3x) - 3|) + 4) - 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2|(-1sin(3\u03c0x + 2) - 2) - 2| + 3) - 3)/(-1(-2|(-1sin(3\u03c0x + 2) - 2) - 2| + 3) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1x^2 + 5x - 1) + 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|((-3x + 1)/(3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(3\u03c0(2((2(5x^2 + 4x + 2))/(-2(5x^2 + 4x + 2) + 3)) - 4) + 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-5((3x + 2)/(3x + 3)) + 5) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3(2x - 2)^2 + 5(2x - 2) + 2) - 3| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((3x - 2)/(3x - 1)) + 1| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(2sin(2\u03c0x) - 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|((-1(-1e^(1(2x^2 + 3x))) - 1)/(-1(-1e^(1(2x^2 + 3x))) + 2)) + 3| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-4((3(3sin(1\u03c0x) + 3) - 2)/(-3(3sin(1\u03c0x) + 3) - 3)) - 5) - 2|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1((-1(-1|x - 2| + 2) + 2)/(-3(-1|x - 2| + 2) + 4)) + 2)^2 + 1(-1((-1(-1|x - 2| + 2) + 2)/(-3(-1|x - 2| + 2) + 4)) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(1\u03c0(2|((1(1x - 1))/(-2(1x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-1(-1|((2x + 2)/(-1x + 2)) + 3|)^2 - 1(-1|((2x + 2)/(-1x + 2)) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1sin(3\u03c0(-1|((3x + 3)/(1x + 3)) - 2| + 1) - 1) + 3)^2 + 3(1sin(3\u03c0(-1|((3x + 3)/(1x + 3)) - 2| + 1) - 1) + 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(1|(2x - 1) - 3| + 2) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0(-2|(-5x^2 + 2x - 2) + 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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-5(-4x + 1)^2 - 3(-4x + 1) - 1) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(3\u03c0(-5(-1|x - 1| + 2)^2 - 1(-1|x - 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) = 1(2|((1(-4x^2 - 2x + 1) - 1)/(-1(-4x^2 - 2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3cos(2\u03c0(2(-1x + 1)^2 + 5(-1x + 1) - 4)) - 3)| - 2 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) = -2|(1(-1((-2x + 3)/(-1x - 2)) + 5)^2 - 4(-1((-2x + 3)/(-1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1|x + 3| + 3)^2 + 5(-1|x + 3| + 3) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((2(-2x - 3) + 2)/(2(-2x - 3) + 1))^2 + 1((2(-2x - 3) + 2)/(2(-2x - 3) + 1)) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(3\u03c0(-4(1|x - 3| + 3) + 5) + 2) - 2 is strictly increasing.\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) = (-1(2cos(1\u03c0x - 3) + 2) + 1)/(-3(2cos(1\u03c0x - 3) + 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|(-4x^2 + 2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(-2((-2x + 3)/(1x - 2)) + 1)) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((3(3|x| - 2))/(-2(3|x| - 2) + 3))^2 - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2e^(1(-4x^2 + 1x + 4))) - 1| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1((2(1|(-1x^2 - 2x + 5) + 2| + 1) + 2)/(-2(1|(-1x^2 - 2x + 5) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3e^(1(-2|((2x + 3)/(-1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(3(3|x| - 2) - 4)^2 + 3(3(3|x| - 2) - 4) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1|(2((3x)/(1x - 2)) + 3) - 3| + 1)^2 - 1(-1|(2((3x)/(1x - 2)) + 3) - 3| + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1|(-3((-3x + 2)/(1x + 1))^2 - 1((-3x + 2)/(1x + 1)) + 4) + 2| - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2sin(3\u03c0x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-1|((-1x - 2)/(2x - 1)) + 2| - 1)^2 - 5(-1|((-1x - 2)/(2x - 1)) + 2| - 1) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-2|x| + 1)^2 - 3(-2|x| + 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1sin(2\u03c0x - 1) + 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(3\u03c0((-3x + 3)/(3x + 3)) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(1(-3x + 5)^2 + 4(-3x + 5))| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(2x + 5) - 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) = -4(4((-1(1|x - 1| - 1) - 1)/(2(1|x - 1| - 1) + 1))^2 - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-1e^(-2((2x - 2)/(2x + 2))) + 2)^2 + 1(-1e^(-2((2x - 2)/(2x + 2))) + 2) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(1\u03c0(-2|x + 1| + 2) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1sin(2\u03c0(-3((1x)/(-1x - 2)) + 3) - 2) + 2)^2 - 2(1sin(2\u03c0(-3((1x)/(-1x - 2)) + 3) - 2) + 2) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0((1x - 2)/(1x - 3)) + 1) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0((3(-2|(4x^2 + 5x - 5)| - 3) - 3)/(-2(-2|(4x^2 + 5x - 5)| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(5(4((2x + 2)/(-3x)) + 2)^2 - 4(4((2x + 2)/(-3x)) + 2) - 3) - 2| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-3cos(1\u03c0((-1x - 3)/(2x)) - 3) + 3) + 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2|(-2sin(2\u03c0x + 2) - 2) + 3| - 1) + 2)/(-1(2|(-2sin(2\u03c0x + 2) - 2) + 3| - 1) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2|(3((-3x - 2)/(1x + 3))^2 + 4((-3x - 2)/(1x + 3)) + 2) - 2| + 1) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-2sin(1\u03c0(1((-1x - 1)/(-3x))^2 - 5) + 2) + 2) + 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2|((-2x + 2)/(2x - 1)) + 2| + 2)^2 + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(5x^2)) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2|x - 3| - 3)^2 - 5(2|x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(2(5x + 4)^2 - 3(5x + 4) - 5)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(2\u03c0((-3(-4x^2 - 1x - 3) + 1)/(-1(-4x^2 - 1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-3cos(1\u03c0(-2(1|x - 3| + 3)^2 - 1(1|x - 3| + 3)) - 2) - 3) - 2)/(1(-3cos(1\u03c0(-2(1|x - 3| + 3)^2 - 1(1|x - 3| + 3)) - 2) - 3) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(-1((3x - 3)/(3x + 3)) + 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": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(1\u03c0(1|(3((2x)/(1x - 2)) - 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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0((1x + 2)/(2x + 1))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(1sin(2\u03c0x + 2) + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1cos(1\u03c0(-1x - 2)) + 3) + 3)/(-1(-1cos(1\u03c0(-1x - 2)) + 3) + 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) = -2sin(1\u03c0(3|x + 3| + 2)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(3cos(2\u03c0(1|x + 1| + 3) + 3) - 1) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(1\u03c0(4(2|x|) + 5)) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(2x^2 - 3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-1cos(1\u03c0x - 3) - 1) + 3)/(1(-1cos(1\u03c0x - 3) - 1) + 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) = (-3(2|(1sin(1\u03c0x) - 3) - 3|) - 1)/(3(2|(1sin(1\u03c0x) - 3) - 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) = 1|(3sin(2\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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3((2(3|x + 2| - 1) - 2)/(2(3|x + 2| - 1) + 3))^2 + 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-2cos(3\u03c0x) + 2) - 3| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1(2|((-2x + 1)/(1x + 1)) + 1| - 1)^2 - 3(2|((-2x + 1)/(1x + 1)) + 1| - 1) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-3(-1|((3x - 2)/(-3x - 3))| + 2)^2 + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0(4(3x - 1)^2 + 4(3x - 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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((1(5(-2|x - 1| + 1)^2 - 3(-2|x - 1| + 1)) - 3)/(-1(5(-2|x - 1| + 1)^2 - 3(-2|x - 1| + 1)) - 2)) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((-2(2x^2 + 4x - 5) - 2)/(-1(2x^2 + 4x - 5) + 2)) - 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-1((1x)/(-1x - 1))^2 + 5((1x)/(-1x - 1))) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "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)^2 + 3(-2sin(3\u03c0x + 1) + 3) - 2) - 3| - 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) = -3cos(3\u03c0(-1|x - 1| + 2) - 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-2((-1(2|x + 3| - 2) + 2)/(3(2|x + 3| - 2) - 2))^2 + 5((-1(2|x + 3| - 2) + 2)/(3(2|x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(1\u03c0(4x) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-1(1x + 2)^2 - 1(1x + 2) + 3) - 2) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(1sin(2\u03c0x) + 2)^2 + 2(1sin(2\u03c0x) + 2) - 5 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) = 3e^(1(-2|(-1(-4x^2 + 5) - 5)| - 2)) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-1|(2((1x)/(1x + 2))^2 - 2) + 3| + 1) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(2((-2(-4x + 5) + 3)/(-2(-4x + 5) - 2))^2 - 3((-2(-4x + 5) + 3)/(-2(-4x + 5) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2|(2cos(1\u03c0((1x + 2)/(3x + 1)) - 3) + 3)| - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-3cos(3\u03c0x + 2) - 1) + 2| + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((1(1e^(1(1|x - 3| - 2)) - 2))/(3(1e^(1(1|x - 3| - 2)) - 2) + 3))^2 - 4((1(1e^(1(1|x - 3| - 2)) - 2))/(3(1e^(1(1|x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2sin(1\u03c0x + 3) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(2|(-2x^2 + 2x + 1) + 2| + 2))/(2(2|(-2x^2 + 2x + 1) + 2| + 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((-1x + 3)/(1x))^2 - 3((-1x + 3)/(1x)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-1(-3|(1x^2 - 2x)|) - 2)/(3(-3|(1x^2 - 2x)|) + 3)) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((-2x + 2)/(-1x - 1))^2 + 5((-2x + 2)/(-1x - 1)) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-1cos(1\u03c0x - 2) - 1) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(-4(1|(-1x^2 + 4x - 1) - 2| - 3) - 2) - 3)/(-2(-4(1|(-1x^2 + 4x - 1) - 2| - 3) - 2) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(2((-3x + 1)/(-2x - 3))^2 - 3) - 2| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1sin(2\u03c0x + 1) + 1)^2 + 1(-1sin(2\u03c0x + 1) + 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(2\u03c0(-1|((3(-3x - 4) - 2)/(1(-3x - 4) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1|(3sin(3\u03c0(-4x^2 - 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": "32", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-3(3|x - 2| - 1) + 4)^2 - 5(-3(3|x - 2| - 1) + 4) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1\u03c0(-2((-3(1x^2 + 3x + 4) + 1)/(1(1x^2 + 3x + 4) - 2)) - 4) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-2|(3sin(1\u03c0((-2x + 3)/(1x + 2)) + 1) - 1) - 2| - 1)^2 + 5 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(-1cos(2\u03c0x) - 1) - 2)/(-3(-1cos(2\u03c0x) - 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) = 2|(2(3((2x + 3)/(3x + 1)) - 5)^2 - 3(3((2x + 3)/(3x + 1)) - 5) + 5) + 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((1(-2|(-2x^2 + 4x - 3) + 2| - 2) - 3)/(3(-2|(-2x^2 + 4x - 3) + 2| - 2) - 2)) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3|(3cos(1\u03c0(3x^2 + 2x - 3) + 2) + 2) + 2| - 2) + 5 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) = (1(-1|(-1(-4x^2 - 5x + 2) - 4) - 2| + 3) + 2)/(-3(-1|(-1(-4x^2 - 5x + 2) - 4) - 2| + 3) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-2(-1sin(3\u03c0x + 2) + 1) - 5) + 3| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-2sin(1\u03c0((3x + 1)/(-2x + 1)) - 2) + 3) - 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1e^(2(-3|(1(3x^2 + 1x - 5) + 2) - 3| + 2)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3cos(1\u03c0((-3x + 3)/(-3x + 4)) + 3))^2 - 1(3cos(1\u03c0((-3x + 3)/(-3x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|((-1(1cos(1\u03c0x) - 1) + 2)/(1(1cos(1\u03c0x) - 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) = -3e^(1(3(1x - 3)^2 + 1(1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(4(-1|(-1cos(2\u03c0x + 3) + 3)| + 1)^2 + 4(-1|(-1cos(2\u03c0x + 3) + 3)| + 1)) - 4 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|(4x^2 + 4x) - 1| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3e^(1(1(2|((1x + 3)/(2x + 1)) + 3|) - 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(4(-1|x + 2| + 3)^2 + 3(-1|x + 2| + 3) + 1) - 2)/(-1(4(-1|x + 2| + 3)^2 + 3(-1|x + 2| + 3) + 1) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1e^(-1(5x^2 + 1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(1\u03c0(-1(-3x^2 + 4x + 4) - 2) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(2sin(2\u03c0(1|(3x + 3) + 2| - 2) + 2) + 1)^2 - 1(2sin(2\u03c0(1|(3x + 3) + 2| - 2) + 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) = -4((1(-1(1|x + 3| - 3) - 1) - 2)/(3(-1(1|x + 3| - 3) - 1) + 1))^2 - 5((1(-1(1|x + 3| - 3) - 1) - 2)/(3(-1(1|x + 3| - 3) - 1) + 1)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-2|(3x - 4) - 3| + 2)^2 + 5(-2|(3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1|(2sin(3\u03c0(-4x^2 - 2x - 2) - 2)) + 2| - 2) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-2((-2(2|x + 1| + 3) + 1)/(2(2|x + 1| + 3) - 3)) - 2)^2 + 5(-2((-2(2|x + 1| + 3) + 1)/(2(2|x + 1| + 3) - 3)) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(1\u03c0(2|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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-3(2((3x - 3)/(-1x + 2))^2 + 4((3x - 3)/(-1x + 2)) - 5)) + 2| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1|(-1x^2 - 3x + 1) - 3| - 3) + 2)/(-1(-1|(-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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(3\u03c0x - 3) + 2) - 3)/(-3(-1sin(3\u03c0x - 3) + 2) + 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) = -1(3cos(2\u03c0((1x - 2)/(-2x + 1)) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((2(-2|(2x + 5)| + 3) - 2)/(2(-2|(2x + 5)| + 3) + 2))^2 - 3((2(-2|(2x + 5)| + 3) - 2)/(2(-2|(2x + 5)| + 3) + 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-3|(-2(-1sin(3\u03c0x + 3) - 2)^2 - 1(-1sin(3\u03c0x + 3) - 2) + 1) + 2| + 1) + 4 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) = 3(-5((1(-3|x - 3| + 3) + 2)/(-3(-3|x - 3| + 3) + 4)) - 4)^2 - 1(-5((1(-3|x - 3| + 3) + 2)/(-3(-3|x - 3| + 3) + 4)) - 4) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(4((-2(-2x + 5) + 2)/(1(-2x + 5)))^2 + 5((-2(-2x + 5) + 2)/(1(-2x + 5))) - 3) + 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(1\u03c0(2(-1|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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((-3x + 1)/(3x + 3))^2 - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0(5((-3(-2|x + 1| - 1) + 1)/(-3(-2|x + 1| - 1) - 1)) - 4) + 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-2(1cos(1\u03c0((3x + 1)/(-3x - 3)) - 2) - 2)^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": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-1|x + 1|)^2 - 1(-1|x + 1|), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2e^(-2(-4((2x)/(1x + 2))^2 + 3((2x)/(1x + 2)) - 1)) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2cos(1\u03c0(5x + 2) - 3) + 2) + 2)/(-2(-2cos(1\u03c0(5x + 2) - 3) + 2) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(4(3x + 2)^2 + 5(3x + 2) - 5) + 2| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-5x^2 + 3x + 5) - 1| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-3sin(2\u03c0x - 2) - 3)^2 + 3(-3sin(2\u03c0x - 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": "28", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(2\u03c0(4(2|(1x + 5) - 1| - 1)^2 + 4(2|(1x + 5) - 1| - 1) + 1) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1cos(3\u03c0(-2x - 3)) + 2) + 3|, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1|(-2(-3x^2 + 5x - 1)) - 2| + 1) + 1)/(1(-1|(-2(-3x^2 + 5x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(5(-3((-2x - 3)/(-3x + 4))^2 + 2((-2x - 3)/(-3x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(5(3|((-2x + 3)/(3x + 3)) - 1|) + 3)^2 - 5(5(3|((-2x + 3)/(3x + 3)) - 1|) + 3) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0((3(-1|x - 1| + 3) + 1)/(-3(-1|x - 1| + 3))) + 3) + 1 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) = 3sin(2\u03c0(-2((2(1x^2 + 5x + 1) + 2)/(2(1x^2 + 5x + 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) = 1(-2cos(2\u03c0(5x^2 - 5x + 5) - 2) + 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3|(2cos(2\u03c0((-2x + 3)/(3x - 2)) + 2) + 3) - 1|) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(2sin(2\u03c0x + 1) - 2) - 2)/(2(2sin(2\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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1|(-1((-1x + 1)/(-2x - 2))^2 - 5((-1x + 1)/(-2x - 2)) + 4)| + 3) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1|(-5(-3x^2 + 5x + 5) - 1) + 2| - 3))/(2(-1|(-5(-3x^2 + 5x + 5) - 1) + 2| - 3) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(-1|(2sin(3\u03c0x - 3) + 1) + 2| + 3)^2 - 5(-1|(2sin(3\u03c0x - 3) + 1) + 2| + 3) + 4) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1cos(3\u03c0(2x^2 + 4x) + 1))| positive?\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(2\u03c0(2|x - 1| + 2) - 1) + 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) = 2sin(2\u03c0(1|(2x^2 + 4x + 3) + 3| + 1) + 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(1|(1((2x - 2)/(1x + 3)) - 1) + 1| + 2)^2 + 4(1|(1((2x - 2)/(1x + 3)) - 1) + 1| + 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1((-3x + 3)/(3x + 2))^2 + 4((-3x + 3)/(3x + 2)) + 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(4(-2sin(3\u03c0x + 3) + 2) + 2)^2 + 1(4(-2sin(3\u03c0x + 3) + 2) + 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2|(-5(1x^2 - 5x - 2) + 2) + 2| + 2) + 1)/(2(-2|(-5(1x^2 - 5x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(1(-4(-3cos(2\u03c0x + 1) - 1) - 1)^2 + 4(-4(-3cos(2\u03c0x + 1) - 1) - 1) + 5) + 1| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(4(2e^(1(3|x - 3| - 1)) - 3)^2 + 2(2e^(1(3|x - 3| - 1)) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1((-1(-2|(-1x^2 - 2x + 5) + 3|) - 3)/(2(-2|(-1x^2 - 2x + 5) + 3|) + 3)) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(-1((-3x + 2)/(-2x - 2)) - 4) + 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-2|(-2((-1x - 1)/(3x - 1))^2 + 1)| + 3) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(2(-3((-1x + 3)/(2x + 2))^2 + 4((-1x + 3)/(2x + 2)) - 3)) + 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2sin(3\u03c0x - 2)) + 2| + 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(-3e^(-2(-5x^2 + 3x - 2)) + 2))/(-1(-3e^(-2(-5x^2 + 3x - 2)) + 2) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1cos(1\u03c0(-2x^2 - 2x) + 1) - 1)| + 3 negative?\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((-1(2(3cos(2\u03c0x - 3))^2 + 4(3cos(2\u03c0x - 3)) - 5) + 1)/(1(2(3cos(2\u03c0x - 3))^2 + 4(3cos(2\u03c0x - 3)) - 5) + 1)) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-1(-2cos(1\u03c0x))^2 + 2(-2cos(1\u03c0x)) + 1) 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(1sin(2\u03c0(1|x + 2| + 3) - 3) - 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-3(3|(-2cos(1\u03c0x + 1) - 2) + 1| + 1) - 2)^2 + 2(-3(3|(-2cos(1\u03c0x + 1) - 2) + 1| + 1) - 2) - 5 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(-1|(5((-3x - 2)/(-3x - 1)) + 1)| - 3)^2 - 2(-1|(5((-3x - 2)/(-3x - 1)) + 1)| - 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((-1x - 2)/(-1x))^2 + 1((-1x - 2)/(-1x)) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(5((3x + 2)/(2x)) + 2) + 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(2sin(1\u03c0(5x - 3) - 3) + 1) + 2| - 3 is strictly decreasing.\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) = 2sin(3\u03c0(3((-2x - 3)/(3x + 1)) + 3) + 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2cos(1\u03c0x) + 1) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-3(-4(-1cos(2\u03c0x - 3) + 1) - 2)^2 + 3(-4(-1cos(2\u03c0x - 3) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(1cos(1\u03c0x + 3) + 2)^2 - 5(1cos(1\u03c0x + 3) + 2) - 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) = -1(1|((3(2x) - 1)/(2(2x) - 3)) + 2| - 2)^2 + 3(1|((3(2x) - 1)/(2(2x) - 3)) + 2| - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-1(-3cos(3\u03c0((-2x)/(-2x + 1)) - 3) - 2)^2 + 4(-3cos(3\u03c0((-2x)/(-2x + 1)) - 3) - 2) - 1) + 3| 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|(2cos(3\u03c0(2x - 5) + 3) - 1) + 1|) - 3)/(-3(1|(2cos(3\u03c0(2x - 5) + 3) - 1) + 1|)), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(1\u03c0(-4(-3|x + 3| + 1) - 1)) + 3 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) = 1(-4((3x)/(-2x - 2)) - 2)^2 - 4(-4((3x)/(-2x - 2)) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((1x - 2)/(-2x + 3))^2 + 5((1x - 2)/(-2x + 3)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-1(-2|x + 1| + 2)^2 - 2(-2|x + 1| + 2) - 5) + 2)/(1(-1(-2|x + 1| + 2)^2 - 2(-2|x + 1| + 2) - 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-3cos(1\u03c0x - 1) - 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) = -2|(-4((-3(5x) + 2)/(1(5x)))^2 - 3((-3(5x) + 2)/(1(5x))) - 2) + 1| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3\u03c0(2|((-3(-2x + 2))/(3(-2x + 2) + 2))| + 2) + 3) - 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(1|((-2x + 1)/(-1x - 2)) - 1| - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-3|((-3x)/(1x + 1)) + 1| - 2)^2 + 5(-3|((-3x)/(1x + 1)) + 1| - 2) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(1|(1sin(2\u03c0x) + 3) + 2| - 2) + 1)/(-1(1|(1sin(2\u03c0x) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-2|(5((-2x + 3)/(2x - 1))^2 - 5((-2x + 3)/(2x - 1))) + 1| - 2) - 1) + 2 is strictly increasing.\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) = -4(-3|(-5x^2 + 5x + 2)| + 2) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((-3(-1(-2|x - 2| - 2) - 3))/(3(-1(-2|x - 2| - 2) - 3) + 3))^2 + 1((-3(-1(-2|x - 2| - 2) - 3))/(3(-1(-2|x - 2| - 2) - 3) + 3)) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(3\u03c0(-2(-4x)^2 - 5(-4x) + 3) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-3sin(3\u03c0(4x^2 + 3x + 2) - 3) - 2) - 3)/(1(-3sin(3\u03c0(4x^2 + 3x + 2) - 3) - 2) + 1) positive?\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((2(-1cos(3\u03c0x - 3) - 3) + 1)/(2(-1cos(3\u03c0x - 3) - 3)))^2 - 3((2(-1cos(3\u03c0x - 3) - 3) + 1)/(2(-1cos(3\u03c0x - 3) - 3))) - 5 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) = -1|(5(-3sin(2\u03c0(-1x + 5)))^2 - 5(-3sin(2\u03c0(-1x + 5))) - 1) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-2|(-1x - 3) + 3|)^2 - 3(-2|(-1x - 3) + 3|) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-3(-5(3sin(2\u03c0x - 3) + 2) + 1)^2 + 5(-5(3sin(2\u03c0x - 3) + 2) + 1) + 5) + 3)/(1(-3(-5(3sin(2\u03c0x - 3) + 2) + 1)^2 + 5(-5(3sin(2\u03c0x - 3) + 2) + 1) + 5) + 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) = (-3(-1sin(2\u03c0(2(-1|x - 3| + 3)^2 - 5(-1|x - 3| + 3) + 2) - 1) - 1) - 3)/(-3(-1sin(2\u03c0(2(-1|x - 3| + 3)^2 - 5(-1|x - 3| + 3) + 2) - 1) - 1) + 2) 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) = -1sin(1\u03c0(-4x + 2) - 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((-3(5(3x - 1)^2 + 2(3x - 1) - 3) - 1)/(-1(5(3x - 1)^2 + 2(3x - 1) - 3) + 2)) - 3| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(3(2|((2x + 1)/(2x - 3)) - 1| - 3)^2 - 5(2|((2x + 1)/(2x - 3)) - 1| - 3) - 4) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((-1(1sin(1\u03c0x - 3) + 2))/(1(1sin(1\u03c0x - 3) + 2) + 2)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1\u03c0(5(2((1x - 2)/(-2x - 1))^2 - 1((1x - 2)/(-2x - 1)) + 5) + 3) + 3) + 2 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) = -4((-3(1cos(1\u03c0x + 3) + 1) + 3)/(-1(1cos(1\u03c0x + 3) + 1) - 2))^2 + 1((-3(1cos(1\u03c0x + 3) + 1) + 3)/(-1(1cos(1\u03c0x + 3) + 1) - 2)) - 5 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((-2(2|(1x) - 3| - 2))/(3(2|(1x) - 3| - 2) - 2))^2 + 5((-2(2|(1x) - 3| - 2))/(3(2|(1x) - 3| - 2) - 2)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((-3(2x + 1) - 1)/(-1(2x + 1) + 2))^2 - 2((-3(2x + 1) - 1)/(-1(2x + 1) + 2)) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-3|x| + 3)^2 + 1(-3|x| + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3(-4x + 5)^2 - 2(-4x + 5) + 1) - 2| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1(2sin(2\u03c0x - 1) + 1)^2 + 2(2sin(2\u03c0x - 1) + 1)) + 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|((1(1cos(3\u03c0x - 3) - 3) - 1)/(-1(1cos(3\u03c0x - 3) - 3) - 1)) - 2| + 3)^2 + 5(-3|((1(1cos(3\u03c0x - 3) - 3) - 1)/(-1(1cos(3\u03c0x - 3) - 3) - 1)) - 2| + 3) - 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) = -2(2|(2sin(1\u03c0(-3x - 1) + 2) + 3) + 1| + 1)^2 - 3(2|(2sin(1\u03c0(-3x - 1) + 2) + 3) + 1| + 1) + 4 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) = -1sin(2\u03c0(2|(2((1x + 1)/(1x + 2)) + 4) + 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": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(3\u03c0((2x + 3)/(-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": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1|x - 3| + 3)^2 - 4(-1|x - 3| + 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1(-2|x + 3|)^2 - 5(-2|x + 3|) + 4) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-2|(2((3x + 2)/(1x - 3)) + 2) + 2| + 3)^2 - 4(-2|(2((3x + 2)/(1x - 3)) + 2) + 2| + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3\u03c0((-3x - 1)/(3x - 1))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1(-2|x - 1| + 3) - 3)^2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1|((2(-3x - 3) - 3)/(-2(-3x - 3) - 1)) + 1| - 1)^2 - 3(1|((2(-3x - 3) - 3)/(-2(-3x - 3) - 1)) + 1| - 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(3|(1x^2 - 3x + 2) - 2| + 3) + 2) + 1 negative?\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) = 1(-2cos(1\u03c0(-3|x - 3| - 1) - 3) - 2)^2 + 3(-2cos(1\u03c0(-3|x - 3| - 1) - 3) - 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(1(2|x - 1| + 3) - 2) + 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) = -5(1sin(3\u03c0x - 3) + 1)^2 - 1(1sin(3\u03c0x - 3) + 1) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-4(3cos(1\u03c0(3x + 2) + 3) - 2)^2 + 2) - 1)/(-2(-4(3cos(1\u03c0(3x + 2) + 3) - 2)^2 + 2) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(-1x^2 + 4x + 5)) + 1 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) = 1cos(3\u03c0(4x - 1) - 2) positive?\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) = -1(-3cos(2\u03c0(-2(2|x - 3| - 2) + 1) + 2) + 3)^2 - 3(-3cos(2\u03c0(-2(2|x - 3| - 2) + 1) + 2) + 3) - 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) = -2|((-1(-4(5x + 2)^2 + 2(5x + 2) + 3) + 2)/(3(-4(5x + 2)^2 + 2(5x + 2) + 3) + 1)) + 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1((1x - 3)/(-1x + 2))^2 + 3((1x - 3)/(-1x + 2)) - 3) - 3| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0((-2(-1x + 2) - 2)/(3(-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": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(1((-2x - 2)/(-3x + 4))^2 - 5((-2x - 2)/(-3x + 4)) - 5) - 2| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(5(-3sin(1\u03c0((-2x - 2)/(2x - 3)) + 1) - 1)^2 - 2(-3sin(1\u03c0((-2x - 2)/(2x - 3)) + 1) - 1) - 4) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(4(1(2|x + 1| - 1)^2 - 2) - 1) - 1)/(1(4(1(2|x + 1| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2sin(3\u03c0((3x - 3)/(2x + 2)) - 2) - 2)^2 - 3(-2sin(3\u03c0((3x - 3)/(2x + 2)) - 2) - 2) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1(4x - 1)^2 - 1(4x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-2sin(3\u03c0x - 1) + 2) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(3\u03c0(1|x - 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": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1|(5((3x + 3)/(3x + 1))) + 3| - 1)^2 + 3(-1|(5((3x + 3)/(3x + 1))) + 3| - 1) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4((-2x)/(3x - 1))^2 - 4((-2x)/(3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(3(-2|((3x + 2)/(-2x + 1)) + 1| + 3)^2 + 3(-2|((3x + 2)/(-2x + 1)) + 1| + 3) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(3|x| - 1)^2 - 2(3|x| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2|x - 1| + 3)^2 + 1(-2|x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-1sin(2\u03c0((-2x - 2)/(-2x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3|(-1sin(1\u03c0(1x^2 - 5x) - 3) + 3) - 1| - 3) - 4 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(1cos(1\u03c0x) + 3)^2 - 1(1cos(1\u03c0x) + 3) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(2sin(1\u03c0(-3|((-3x + 2)/(-2x + 3)) + 3| - 2) - 3) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0((-1x + 3)/(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": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(1\u03c0(-3|((-1x)/(-2x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1cos(1\u03c0x) + 2) - 3| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1|((3x + 3)/(1x)) - 1| - 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(3|(-5(-1cos(1\u03c0x + 1) + 1) - 3) - 2| - 2)^2 - 2(3|(-5(-1cos(1\u03c0x + 1) + 1) - 3) - 2| - 2) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(3\u03c0(5(3(-2|x - 2| - 1)^2 + 2) - 3) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1((2x + 3)/(2x - 3))^2 - 3((2x + 3)/(2x - 3)) - 5) - 2| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-1sin(2\u03c0x - 2) - 3) - 3)/(2(-1sin(2\u03c0x - 2) - 3) - 1) 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(-3sin(1\u03c0(3(-1|x + 2| + 1) - 5) - 2) + 2)^2 - 3(-3sin(1\u03c0(3(-1|x + 2| + 1) - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1cos(1\u03c0x - 1)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(3\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": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3|(-2cos(2\u03c0x - 1) + 3) + 1| + 3)^2 - 3(-3|(-2cos(2\u03c0x - 1) + 3) + 1| + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2|x - 3| - 3)^2 + 5(2|x - 3| - 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3cos(3\u03c0(1(3|x + 1| - 1) - 2) + 3) - 1))/(1(-3cos(3\u03c0(1(3|x + 1| - 1) - 2) + 3) - 1) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3\u03c0((1(-2x^2 - 4x - 3) - 1)/(2(-2x^2 - 4x - 3) + 1)) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((-2x - 2)/(-3x)) + 2| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(1\u03c0(-5x^2 - 5x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-3cos(2\u03c0((-3x)/(-3x - 3)) + 2) + 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) = 1|(4(3((-1x + 1)/(1x + 3))^2 + 5((-1x + 1)/(1x + 3)) + 1) - 2) - 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(3cos(1\u03c0x) - 3) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(3cos(1\u03c0x - 1) - 2)^2 + 1(3cos(1\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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((-2x + 2)/(3x + 2))^2 + 1((-2x + 2)/(3x + 2)) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((-2(2|(1x + 5) + 1| - 3) - 1)/(-2(2|(1x + 5) + 1| - 3) + 3))^2 - 3((-2(2|(1x + 5) + 1| - 3) - 1)/(-2(2|(1x + 5) + 1| - 3) + 3)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-3(-2sin(1\u03c0(3x - 5) + 3) - 2)^2 - 3(-2sin(1\u03c0(3x - 5) + 3) - 2) - 5) + 2)/(3(-3(-2sin(1\u03c0(3x - 5) + 3) - 2)^2 - 3(-2sin(1\u03c0(3x - 5) + 3) - 2) - 5) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3sin(1\u03c0(1x^2 + 1x + 5) + 2)) + 3)/(3(-3sin(1\u03c0(1x^2 + 1x + 5) + 2)) + 2) 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) = 4((2(-3e^(1(-3x^2 + 3x - 3)) + 1) - 3)/(2(-3e^(1(-3x^2 + 3x - 3)) + 1) + 1)) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-2(-2|x + 1|) + 2)/(3(-2|x + 1|)))^2 - 2((-2(-2|x + 1|) + 2)/(3(-2|x + 1|))) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1((-1x - 3)/(3x + 2)) - 2)^2 + 4(-1((-1x - 3)/(3x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1\u03c0(5x^2 + 2) - 3) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-5((-1x + 1)/(2x)) - 4)^2 - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((2(3|(1x - 1) - 1| - 2) - 3)/(-3(3|(1x - 1) - 1| - 2) + 1))^2 + 2((2(3|(1x - 1) - 1| - 2) - 3)/(-3(3|(1x - 1) - 1| - 2) + 1)) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2\u03c0(2(3x^2 - 2) - 3) + 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-5(-4((1x + 2)/(-1x + 2))^2 - 4((1x + 2)/(-1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3e^(-1(2|(5(-3x + 1)^2 - 3(-3x + 1)) - 3| - 3)) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(-1cos(2\u03c0(-4x^2 - 1x + 5) + 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": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-5(1cos(3\u03c0x + 2) + 3)^2 + 3(1cos(3\u03c0x + 2) + 3) + 1) + 1)/(-1(-5(1cos(3\u03c0x + 2) + 3)^2 + 3(1cos(3\u03c0x + 2) + 3) + 1) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1\u03c0((-2(-2|x + 1| + 1) + 1)/(-3(-2|x + 1| + 1) - 1)) - 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) = (-3(5(-4(-1|x|) - 1)^2 + 4(-4(-1|x|) - 1) + 2) + 1)/(1(5(-4(-1|x|) - 1)^2 + 4(-4(-1|x|) - 1) + 2) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3|(-5x^2 + 3x + 4) - 1| + 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(1\u03c0((-3x - 2)/(2x + 2)) + 1) + 2 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) = -2sin(1\u03c0(-1(-4x - 1)^2 + 5(-4x - 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": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(4((1(3|x + 2| + 2))/(-2(3|x + 2| + 2) + 3)) + 5)^2 + 5(4((1(3|x + 2| + 2))/(-2(3|x + 2| + 2) + 3)) + 5) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2cos(2\u03c0(2x^2 + 1x - 4) + 1) - 1))/(2(2cos(2\u03c0(2x^2 + 1x - 4) + 1) - 1) + 1) 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) = -5(3|((-3x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-5((2x + 2)/(-3x + 4))^2 - 1((2x + 2)/(-3x + 4)) - 2) - 1| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-3cos(3\u03c0x + 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": "For the function f(x) = 2cos(1\u03c0(1|(3(-1x + 4)^2 - 4(-1x + 4) + 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": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(5((1(-2|x - 2| + 1) + 3)/(2(-2|x - 2| + 1) + 1)))^2 - 2(5((1(-2|x - 2| + 1) + 3)/(2(-2|x - 2| + 1) + 1))) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0((1x - 3)/(-2x + 3)) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1(5((1x + 1)/(-2x + 3)) - 4)^2 - 2) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(4(-1(2cos(1\u03c0x - 3)) - 2)^2 - 3(-1(2cos(1\u03c0x - 3)) - 2) + 3) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-2e^(-1(-1(3x^2 + 2x - 4) - 5)) + 1) - 3)/(-1(-2e^(-1(-1(3x^2 + 2x - 4) - 5)) + 1) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((-2x - 2)/(1x)) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2(-1(1cos(1\u03c0x - 1) + 3)^2 - 5(1cos(1\u03c0x - 1) + 3) - 4) + 1) + 3)/(-2(2(-1(1cos(1\u03c0x - 1) + 3)^2 - 5(1cos(1\u03c0x - 1) + 3) - 4) + 1) - 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((2(-5x^2 - 5x - 3) + 3)/(-1(-5x^2 - 5x - 3) - 3)) + 2| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2|x + 1|)^2 - 3(2|x + 1|) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2sin(3\u03c0((2(-1x) + 2)/(-2(-1x) - 3))) - 3)^2 - 4(-2sin(3\u03c0((2(-1x) + 2)/(-2(-1x) - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(3(-3(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) - 4) - 2) - 2)/(1(3(-3(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) - 4) - 2) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(2\u03c0((-2x)/(-3x + 4)) - 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(3sin(2\u03c0(-1(2|x - 1| + 2) - 2) + 3) - 2)^2 + 1(3sin(2\u03c0(-1(2|x - 1| + 2) - 2) + 3) - 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|((2(3(-5x)^2 + 1) + 2)/(3(3(-5x)^2 + 1) - 2)) - 1| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(3(-2|x| + 2) + 5)^2 + 1(3(-2|x| + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2x^2 + 3x + 3) + 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(2(-1|x + 2| + 3)^2 - 4(-1|x + 2| + 3) + 4) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1cos(2\u03c0(-4x - 4) + 2) + 1))/(-3(-1cos(2\u03c0(-4x - 4) + 2) + 1) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1e^(1(3|(2x^2 - 3x - 2) - 1| - 2)) - 3))/(2(-1e^(1(3|(2x^2 - 3x - 2) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(1\u03c0(-2|x + 1|)) - 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) = 5(1sin(2\u03c0x + 1) + 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1((1(1cos(3\u03c0x - 2) + 2) - 1)/(3(1cos(3\u03c0x - 2) + 2) + 3)) + 2)| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(3cos(2\u03c0(-5(1x^2 - 2x - 4) + 3) + 2) - 3)| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1|(2(5x^2 - 4x) + 1) - 1| - 3) - 2)/(2(-1|(2(5x^2 - 4x) + 1) - 1| - 3) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3cos(1\u03c0((-2x + 3)/(-2x + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-1|((-2(-1x^2 + 5x - 4) + 2)/(-1(-1x^2 + 5x - 4) - 1)) - 3| + 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(1cos(3\u03c0(2|((-1x - 2)/(3x + 3)) + 3| - 3) - 3) + 3) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-2x^2 + 5x + 5) - 2| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(2\u03c0(-2x^2 - 5x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(-1((-3(2x^2 + 1x - 2) + 1)/(-1(2x^2 + 1x - 2) - 2)) + 5) + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3cos(2\u03c0x) + 2)^2 - 5(-3cos(2\u03c0x) + 2) + 4 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) = 4(-3|((2(-1x^2 - 5x + 1))/(2(-1x^2 - 5x + 1) - 1)) - 1| - 3) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(-4(-2sin(2\u03c0((-3x)/(2x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(2cos(1\u03c0(2(-1x + 2)^2 - 4(-1x + 2) - 5) + 3) - 2))/(-1(2cos(1\u03c0(2(-1x + 2)^2 - 4(-1x + 2) - 5) + 3) - 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(2|(-4((2x - 3)/(2x + 3))^2 + 3((2x - 3)/(2x + 3)) - 1) + 1| - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(5((-1(3|x + 2| - 3) - 3)/(1(3|x + 2| - 3) + 2)) - 1)^2 - 5(5((-1(3|x + 2| - 3) - 3)/(1(3|x + 2| - 3) + 2)) - 1) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(3|(2cos(3\u03c0x + 2) + 1) - 2| - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(5(4x^2 - 4x - 3) + 1) - 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2\u03c0(-3|x + 1| + 1) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-5(1x + 3)^2 - 5(1x + 3)) - 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3|(-1(4x - 4)^2 - 5(4x - 4) - 4) - 1| + 2) + 2)/(1(3|(-1(4x - 4)^2 - 5(4x - 4) - 4) - 1| + 2)) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((-1(1cos(3\u03c0x - 1) - 2) + 1)/(-2(1cos(3\u03c0x - 1) - 2) + 3))^2 + 1((-1(1cos(3\u03c0x - 1) - 2) + 1)/(-2(1cos(3\u03c0x - 1) - 2) + 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) = -2(-3((1x + 1)/(2x))^2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-4(-1|((2x - 1)/(3x + 3))|) - 5)^2 + 3(-4(-1|((2x - 1)/(3x + 3))|) - 5) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-1sin(2\u03c0x - 3) - 1))/(2(-1sin(2\u03c0x - 3) - 1) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((1(3|(-1x^2 + 3x)| - 2) + 3)/(-2(3|(-1x^2 + 3x)| - 2) - 2)) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(4(2cos(1\u03c0(-1x + 5) + 2) - 3)^2 + 1(2cos(1\u03c0(-1x + 5) + 2) - 3) - 4) - 2)/(2(4(2cos(1\u03c0(-1x + 5) + 2) - 3)^2 + 1(2cos(1\u03c0(-1x + 5) + 2) - 3) - 4) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1sin(3\u03c0x + 3) + 1) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2sin(1\u03c0(2x^2 + 5x + 4) - 2) + 3) - 3)/(-2(-2sin(1\u03c0(2x^2 + 5x + 4) - 2) + 3) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1\u03c0(-2x - 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) = 2(1cos(2\u03c0x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-2|(-2((1x - 3)/(2x - 1))^2 - 1((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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2((2(1cos(3\u03c0(2x^2 - 5x - 3) + 1) + 3) - 3)/(2(1cos(3\u03c0(2x^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": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3(-1sin(1\u03c0x + 2))^2 + 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(1((3(5x - 5) - 1)/(2(5x - 5) - 1))^2 - 2((3(5x - 5) - 1)/(2(5x - 5) - 1)) + 5)| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2sin(1\u03c0x + 1) + 3)^2 + 2(-2sin(1\u03c0x + 1) + 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-1cos(1\u03c0(2x + 1) + 3) - 3))/(3(-1cos(1\u03c0(2x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((3(-5x^2 + 1x + 2) + 2)/(-2(-5x^2 + 1x + 2) + 3)) + 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "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| - 3)^2 + 2(-1|(-3cos(3\u03c0x + 3)) + 1| - 3) - 5 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(2|(3((-2x + 3)/(-2x))^2 + 2((-2x + 3)/(-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": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1cos(3\u03c0x) - 3) + 1| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(2|((-3(3cos(1\u03c0x - 1) + 3) + 2)/(2(3cos(1\u03c0x - 1) + 3) - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-5((-1(1cos(3\u03c0x + 1) - 3) + 2)/(-1(1cos(3\u03c0x + 1) - 3)))^2 + 5((-1(1cos(3\u03c0x + 1) - 3) + 2)/(-1(1cos(3\u03c0x + 1) - 3))) - 1) - 5 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(-2(-3sin(3\u03c0(2|x + 3| - 1) + 2) + 1)^2 + 2(-3sin(3\u03c0(2|x + 3| - 1) + 2) + 1) + 4) - 3)/(1(-2(-3sin(3\u03c0(2|x + 3| - 1) + 2) + 1)^2 + 2(-3sin(3\u03c0(2|x + 3| - 1) + 2) + 1) + 4) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(5(2(1|x| - 3) - 1)^2 - 3(2(1|x| - 3) - 1)) - 2)/(-2(5(2(1|x| - 3) - 1)^2 - 3(2(1|x| - 3) - 1))) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(-1sin(1\u03c0x - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1cos(2\u03c0x + 2) - 2) - 2| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3\u03c0(1|x + 2| - 1) - 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(4(-1|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": "81", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(-4(-2|x| + 3) + 5) - 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0(3(-2|(-5x - 2) - 3|)^2 + 3(-2|(-5x - 2) - 3|) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-1cos(1\u03c0((-2x + 3)/(3x)) + 1) + 2) + 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) = (-3(2|(3(1x + 4)^2 - 2(1x + 4) - 5) - 1| + 1) - 3)/(3(2|(3(1x + 4)^2 - 2(1x + 4) - 5) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1e^(-1(-2|(-5x^2 - 4x + 3) + 1| - 3)) + 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3sin(1\u03c0(-2|x + 3| - 1) - 1) - 1)^2 + 4(-3sin(1\u03c0(-2|x + 3| - 1) - 1) - 1) + 5 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) = 1(-1sin(2\u03c0((-2x)/(-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": "12", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(1|(-4x + 1)| + 1) + 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(3(2sin(2\u03c0((-3x + 2)/(-2x - 3)) - 2) + 2)^2 - 1(2sin(2\u03c0((-3x + 2)/(-2x - 3)) - 2) + 2) + 3) - 2|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1\u03c0(-2(1|((1x + 1)/(-1x)) - 1| - 1)^2 - 2(1|((1x + 1)/(-1x)) - 1| - 1) + 3) - 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(1x - 1) + 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) = -1sin(1\u03c0(1|x + 2| - 3)) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2(-2sin(1\u03c0((-3x - 3)/(-1x + 2)) + 3) + 2)^2 + 1(-2sin(1\u03c0((-3x - 3)/(-1x + 2)) + 3) + 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) = -1|(-1cos(2\u03c0x - 2) + 3) + 2| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1(5(-2|x| + 2) + 3)^2 + 2(5(-2|x| + 2) + 3) + 4) - 2)/(-3(1(5(-2|x| + 2) + 3)^2 + 2(5(-2|x| + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(-5(-2|(2x^2 - 2) - 1|) - 2) - 3)/(-2(-5(-2|(2x^2 - 2) - 1|) - 2) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3\u03c0(-1x^2 + 5x + 3) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(1|(1sin(2\u03c0x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-1sin(3\u03c0x + 3) + 3) - 3)/(-1(-1sin(3\u03c0x + 3) + 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) = (-2(3|(-2sin(2\u03c0(4x) + 1) + 1) - 3| - 3) + 1)/(1(3|(-2sin(2\u03c0(4x) + 1) + 1) - 3| - 3) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(2\u03c0(1(-2x - 3)^2 + 4(-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": "44", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5((-1(-1(-2sin(1\u03c0x + 2) - 3)) + 2)/(-1(-1(-2sin(1\u03c0x + 2) - 3)) - 1))^2 - 2((-1(-1(-2sin(1\u03c0x + 2) - 3)) + 2)/(-1(-1(-2sin(1\u03c0x + 2) - 3)) - 1)) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-3((3x)/(1x + 2))^2 + 1((3x)/(1x + 2)) - 1) + 1) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-5(2cos(3\u03c0x - 2) + 1)^2 - 4(2cos(3\u03c0x - 2) + 1)) - 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(3\u03c0(-2(2(-3|x - 1| - 1))^2 - 1(2(-3|x - 1| - 1)) + 4) + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(1(1cos(1\u03c0x - 3) - 3)^2 - 4(1cos(1\u03c0x - 3) - 3) - 3))/(-2(1(1cos(1\u03c0x - 3) - 3)^2 - 4(1cos(1\u03c0x - 3) - 3) - 3) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1((-3(-3x - 4) - 1)/(1(-3x - 4) - 3))^2 - 5((-3(-3x - 4) - 1)/(1(-3x - 4) - 3)) + 5) + 1| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3|(-5x^2 - 5x + 5) + 1| + 2) + 2)/(2(3|(-5x^2 - 5x + 5) + 1| + 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(1cos(2\u03c0(5x + 4) + 1) - 1))/(2(1cos(2\u03c0(5x + 4) + 1) - 1) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(1\u03c0(-1|(1(-5x^2 + 1x - 4) + 3) - 1| + 3) + 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(1(2|((1x - 1)/(-2x + 1)) + 1| + 2)^2 - 3(2|((1x - 1)/(-2x + 1)) + 1| + 2) + 1) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(3|((-2x - 1)/(1x - 3)) - 2| - 2)^2 - 4(3|((-2x - 1)/(1x - 3)) - 2| - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(3\u03c0(4(-1((-2x - 2)/(2x - 1))^2 + 4((-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": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(2(1cos(3\u03c0(-3x - 1) + 1) - 1)^2 - 3) - 1)/(-3(2(1cos(3\u03c0(-3x - 1) + 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": "87", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(2sin(1\u03c0x - 2) - 1) - 3|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3|(-1(-1x^2 + 5x + 1) + 4) - 1| - 2) - 2)/(-1(-3|(-1(-1x^2 + 5x + 1) + 4) - 1| - 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(3sin(2\u03c0((1x - 1)/(3x + 3)) - 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) = -3|(-1((1(2cos(1\u03c0x + 3) + 1) + 2)/(2(2cos(1\u03c0x + 3) + 1) - 3))^2 + 3((1(2cos(1\u03c0x + 3) + 1) + 2)/(2(2cos(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": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2|(2((-1x + 1)/(1x))^2 + 1((-1x + 1)/(1x)) + 2) - 3| - 2) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(3|(-2cos(1\u03c0(-3x + 1) + 2) + 3)|)^2 + 2(3|(-2cos(1\u03c0(-3x + 1) + 2) + 3)|) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3(1x + 2)^2 - 3)| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(3sin(2\u03c0((-1x + 1)/(1x)) - 2) + 2) + 2| - 3 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) = -2|(3cos(3\u03c0(-1x^2 - 2x - 1) - 3) + 1) - 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) = 3|((-2(-4(3sin(1\u03c0x + 2) - 2) - 3))/(-3(-4(3sin(1\u03c0x + 2) - 2) - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(4x^2 - 1x - 1)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(3(2cos(1\u03c0(2|x - 1| + 2) + 3) + 2) + 2)^2 + 1(3(2cos(1\u03c0(2|x - 1| + 2) + 3) + 2) + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3\u03c0(2x^2 - 3x - 1) - 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0((-2x)/(-3x + 1)) - 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3(-3sin(2\u03c0x + 3))^2 + 5(-3sin(2\u03c0x + 3)) - 3) + 2| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((2(-1(2|x + 3| - 3) + 3) - 1)/(-2(-1(2|x + 3| - 3) + 3) - 3))^2 + 1((2(-1(2|x + 3| - 3) + 3) - 1)/(-2(-1(2|x + 3| - 3) + 3) - 3)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(5((2(2|x - 1|) - 3)/(1(2|x - 1|)))^2 - 1((2(2|x - 1|) - 3)/(1(2|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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0(-3|(-1x^2 - 2) + 1| - 1) - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(2cos(3\u03c0x + 3) + 3) - 3| + 2)^2 - 4(-1|(2cos(3\u03c0x + 3) + 3) - 3| + 2) - 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) = 3|(-4(-1sin(1\u03c0x - 3) - 3) - 2)| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3((3(-5x - 1) + 3)/(1(-5x - 1) - 3))^2 - 2((3(-5x - 1) + 3)/(1(-5x - 1) - 3))) + 2| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-5((-2x + 1)/(-3x + 4))^2 + 2((-2x + 1)/(-3x + 4)) + 3) - 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3cos(3\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": "90", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(1\u03c0(5x^2 - 3x) + 3) positive?\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(1cos(1\u03c0x - 3) - 3) - 3)/(-1(1cos(1\u03c0x - 3) - 3) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-2sin(2\u03c0x + 2) + 3)^2 + 4(-2sin(2\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(-3x^2 + 5x) + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-2((2(2x + 3) + 2)/(-2(2x + 3) + 3))^2 - 5((2(2x + 3) + 2)/(-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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(3cos(2\u03c0x - 2) - 1) + 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1e^(2(-2(4x + 2)^2 - 2(4x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-2(-2|(3sin(3\u03c0(2x^2 - 2x + 4) + 1) + 1) + 3| + 3) + 2)/(-1(-2|(3sin(3\u03c0(2x^2 - 2x + 4) + 1) + 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": "46", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(3sin(3\u03c0(2|x + 3| - 3) - 3) + 2)^2 + 3(3sin(3\u03c0(2|x + 3| - 3) - 3) + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(-1|((3(-1x^2 + 4x) - 3)/(-1(-1x^2 + 4x))) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(4(-2sin(1\u03c0(3|x - 2| - 3) - 2) + 2)^2 + 5(-2sin(1\u03c0(3|x - 2| - 3) - 2) + 2) + 4))/(3(4(-2sin(1\u03c0(3|x - 2| - 3) - 2) + 2)^2 + 5(-2sin(1\u03c0(3|x - 2| - 3) - 2) + 2) + 4) + 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) = -1|(-1cos(2\u03c0((-3(5x - 1) - 1)/(3(5x - 1) - 2)) + 3)) + 2| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(1e^(-2(-3|x - 1| - 3))) - 1)/(3(1e^(-2(-3|x - 1| - 3))) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-1cos(2\u03c0((-1(-2|x + 3| + 2) - 3)/(1(-2|x + 3| + 2) + 1)) - 2) - 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(2\u03c0(-2|x| + 2) - 3) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-2(2|(-4x^2 - 2x + 4) - 2| + 2) - 1) - 2)/(2(-2(2|(-4x^2 - 2x + 4) - 2| + 2) - 1) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-2|((-3(-1x + 2) + 2)/(2(-1x + 2) - 1)) - 2| - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-3(1cos(2\u03c0(1x^2 + 3x + 2) + 3) + 1) - 3)/(-1(1cos(2\u03c0(1x^2 + 3x + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(-2cos(3\u03c0(5x^2 - 2x + 5) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-1cos(3\u03c0(-2|(-1x + 4) + 2|) - 1) - 3)^2 + 4(-1cos(3\u03c0(-2|(-1x + 4) + 2|) - 1) - 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(-1(2|x + 3| + 3) - 1) - 2) - 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) = 1|(5x^2 - 1x - 4) - 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2cos(3\u03c0(1|x - 3| - 1) + 3) - 1) + 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|(-4(-4x + 5)^2 - 2(-4x + 5))| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2sin(2\u03c0x) - 1) + 3)/(3(-2sin(2\u03c0x) - 1) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(2(-3cos(2\u03c0(5x^2) - 3)) + 4) + 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3((-3(3sin(3\u03c0x - 1) - 3) + 1)/(2(3sin(3\u03c0x - 1) - 3)))^2 - 5((-3(3sin(3\u03c0x - 1) - 3) + 1)/(2(3sin(3\u03c0x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(4(-3((-1x)/(2x + 3)) + 3)^2 - 1(-3((-1x)/(2x + 3)) + 3) - 2) - 1| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((-3(-2|x - 3|) + 2)/(2(-2|x - 3|) + 3))^2 + 4((-3(-2|x - 3|) + 2)/(2(-2|x - 3|) + 3)) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-1(-3((2x + 3)/(-2x + 3))^2 + 3((2x + 3)/(-2x + 3)) + 5) + 4) - 3| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-3(3sin(2\u03c0x + 3) - 1) + 1) + 3)/(-2(-3(3sin(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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1|(-3cos(3\u03c0(4x^2 - 3x) - 3) + 1) - 3| - 3) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|((2x - 1)/(-1x + 2)) - 1| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((2(1cos(1\u03c0(-1x + 1) + 2) - 3) - 1)/(1(1cos(1\u03c0(-1x + 1) + 2) - 3) - 2)) + 3| - 3 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) = 5(-3|((-3x - 2)/(-3x)) - 3|)^2 - 4(-3|((-3x - 2)/(-3x)) - 3|) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1\u03c0(-3((3(-1x^2 - 5x + 1) + 2)/(3(-1x^2 - 5x + 1) - 1)) - 5) + 3) - 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) = -3sin(3\u03c0(5(-3|((-1x + 1)/(3x - 2)) - 1| + 1)^2 + 5(-3|((-1x + 1)/(3x - 2)) - 1| + 1) - 3)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((3(-1|x + 2|) + 2)/(-3(-1|x + 2|) - 3))^2 + 5((3(-1|x + 2|) + 2)/(-3(-1|x + 2|) - 3)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-2sin(3\u03c0x - 3)) + 2)/(2(-2sin(3\u03c0x - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2\u03c0(-3|(1x + 2) + 1| - 2) - 3) - 1 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) = -1(-3|(1sin(1\u03c0x - 1) + 2) + 1| + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-3cos(1\u03c0(1|x + 2| - 3) - 1) - 2) + 2)/(-1(-3cos(1\u03c0(1|x + 2| - 3) - 1) - 2) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2\u03c0(-4((-2(2x^2 + 4x) + 2)/(-1(2x^2 + 4x) + 2)) + 4) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2|(-2x^2 + 4x - 3) + 2| + 3) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((1(-2e^(1(4x - 4)) + 1) - 2)/(2(-2e^(1(4x - 4)) + 1) + 2)) - 1| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(3sin(3\u03c0x) - 3)| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3cos(1\u03c0(1|x|) - 1) - 3) - 2)/(3(-3cos(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) = -2(5(-2cos(1\u03c0(-3|x| + 1) + 1) - 2)^2 + 5) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(1cos(1\u03c0x) + 2)^2 + 3(1cos(1\u03c0x) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((1x - 2)/(3x - 2))| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2cos(2\u03c0x - 2) - 1) - 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) = 3|(3x^2 + 5x - 5) + 1| + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2\u03c0((-1(-1(5x + 2)^2 - 3(5x + 2) + 5) + 1)/(1(-1(5x + 2)^2 - 3(5x + 2) + 5) - 2)) + 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(3cos(2\u03c0x + 1) + 3)^2 + 2(3cos(2\u03c0x + 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) = 3((-3(1(-1|x + 1| + 2) + 1) - 3)/(-1(1(-1|x + 1| + 2) + 1) - 2))^2 - 5((-3(1(-1|x + 1| + 2) + 1) - 3)/(-1(1(-1|x + 1| + 2) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(1(2sin(1\u03c0((-2x + 2)/(-1x))) - 3)^2 + 1(2sin(1\u03c0((-2x + 2)/(-1x))) - 3) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-4(-1x^2 - 4x - 1) + 2) + 3| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(1sin(2\u03c0(1((3x + 1)/(-1x - 1)) + 2) + 2) + 1)^2 - 3 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) = -1sin(2\u03c0(2|((-1(5x - 3) - 3)/(-3(5x - 3) - 3)) + 2| + 3) + 2) + 3 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) = (1(-2|(5(3cos(3\u03c0x - 3) + 2)^2 + 4(3cos(3\u03c0x - 3) + 2) - 3) + 1| + 1) - 1)/(-2(-2|(5(3cos(3\u03c0x - 3) + 2)^2 + 4(3cos(3\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": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3|(3sin(2\u03c0(5x + 4) + 2) - 2) - 3| - 1) + 3)/(2(3|(3sin(2\u03c0(5x + 4) + 2) - 2) - 3| - 1) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(5(3cos(3\u03c0x + 1) + 3)^2 + 1(3cos(3\u03c0x + 1) + 3) - 4) + 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|((1(1cos(3\u03c0x + 3) + 1) - 2)/(-2(1cos(3\u03c0x + 3) + 1) - 1)) - 3| - 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) = (3(1cos(3\u03c0x - 1)) - 3)/(-3(1cos(3\u03c0x - 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) = 2(-2sin(2\u03c0x + 2))^2 - 2(-2sin(2\u03c0x + 2)) + 5 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) = 2cos(3\u03c0(3|x - 3| + 2) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|((-3x)/(2x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3x - 1)/(-3x + 4))^2 + 5((-3x - 1)/(-3x + 4)) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3e^(-2(-3(-2|x + 1| + 3) - 3)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(2|((-1x + 1)/(-3x - 1)) + 3| - 2) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-5(-2sin(1\u03c0((1x - 3)/(-1x - 1))) + 3)^2 + 2(-2sin(1\u03c0((1x - 3)/(-1x - 1))) + 3) + 1) - 1| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1|(1sin(3\u03c0x - 3) - 2) + 3| - 3) + 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(5(2|((-1x - 3)/(-2x + 3)) - 1| - 1))^2 - 1(5(2|((-1x - 3)/(-2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(-1cos(2\u03c0(-4((-1x)/(-2x + 3)) + 4) - 3) + 3)^2 - 5(-1cos(2\u03c0(-4((-1x)/(-2x + 3)) + 4) - 3) + 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(1sin(2\u03c0(-3(3x^2 + 3) + 2) + 3)) + 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 5(-3sin(2\u03c0x + 1) + 2)^2 + 2(-3sin(2\u03c0x + 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) = -1|((1(1cos(2\u03c0(4x - 1) - 2) + 1) + 3)/(-2(1cos(2\u03c0(4x - 1) - 2) + 1))) - 3| - 1 positive?\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|(2cos(1\u03c0(-1x^2 - 4x + 3) + 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": "49", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(2cos(3\u03c0(1|(1x - 2) - 1| + 3) - 3) + 1)^2 - 1(2cos(3\u03c0(1|(1x - 2) - 1| + 3) - 3) + 1) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(3sin(3\u03c0x - 1) + 2)^2 + 1(3sin(3\u03c0x - 1) + 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(1\u03c0((-1(-1|x|) - 3)/(2(-1|x|) - 3))) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3e^(1(-3(-1|x - 1| + 2)^2 + 3(-1|x - 1| + 2) - 1)) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(1((3(-1x) - 1)/(3(-1x) + 3))^2 - 2((3(-1x) - 1)/(3(-1x) + 3)) + 2) - 2| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "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| - 3) + 3) - 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(1(-3x - 5)^2 - 3(-3x - 5) - 1) + 2| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4((-3(2sin(3\u03c0(2x + 2)) + 2) - 1)/(-1(2sin(3\u03c0(2x + 2)) + 2) + 2))^2 - 4((-3(2sin(3\u03c0(2x + 2)) + 2) - 1)/(-1(2sin(3\u03c0(2x + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3\u03c0(4x^2 + 3x - 3) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(-1e^(-1(5x^2 - 1x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-2sin(1\u03c0x + 1) - 2) - 2| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(2((-2(3|x - 3| + 3) + 3)/(2(3|x - 3| + 3) + 3))^2 + 3((-2(3|x - 3| + 3) + 3)/(2(3|x - 3| + 3) + 3)) + 2) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(2cos(2\u03c0(-3|x + 2|) - 2) - 1)^2 - 5(2cos(2\u03c0(-3|x + 2|) - 2) - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(1\u03c0(-3x^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": "59", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((1(4x + 3) - 3)/(-2(4x + 3) + 3))| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3sin(3\u03c0(-3|(-5x^2 - 2x + 2) + 1| + 1) + 3)) - 3)/(1(3sin(3\u03c0(-3|(-5x^2 - 2x + 2) + 1| + 1) + 3)) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2e^(1(-1(-2|x + 2| - 2)^2 + 2(-2|x + 2| - 2) + 2)) - 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(3sin(3\u03c0(-2|(3x - 4) - 3| - 3) - 3) + 2)^2 - 1(3sin(3\u03c0(-2|(3x - 4) - 3| - 3) - 3) + 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-3cos(1\u03c0(2|x + 1| - 1) + 1) + 2) + 2)/(-2(-3cos(1\u03c0(2|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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(1\u03c0((1x - 1)/(-2x - 1)) - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2(2sin(3\u03c0(5(3|x + 3| - 2)^2 - 1(3|x + 3| - 2) - 5)) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2(-1|x + 3| + 1)^2 - 3(-1|x + 3| + 1) + 1) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(3\u03c0(1x + 3) + 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) = (-2(-3sin(2\u03c0(-2x - 4) + 2)) - 2)/(-2(-3sin(2\u03c0(-2x - 4) + 2)) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2cos(3\u03c0(5x^2 + 1x + 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": "42", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2sin(1\u03c0(3|((2x - 2)/(-2x)) + 2| + 1) - 3) + 2)^2 - 4(2sin(1\u03c0(3|((2x - 2)/(-2x)) + 2| + 1) - 3) + 2) + 4 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|(3(5((-1x + 3)/(3x + 1)) - 4)^2 + 1(5((-1x + 3)/(3x + 1)) - 4) - 3) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((-3(3|(3x + 5)| - 1) - 2)/(-2(3|(3x + 5)| - 1) + 3))^2 - 4((-3(3|(3x + 5)| - 1) - 2)/(-2(3|(3x + 5)| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1\u03c0(3(-5((-2x - 3)/(1x))^2 - 1((-2x - 3)/(1x))) + 5) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1sin(1\u03c0((-1x + 2)/(-2x + 1)) - 1) + 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(2\u03c0x + 3) + 1) + 3)/(-1(-1sin(2\u03c0x + 3) + 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0((2(-5x + 1) + 2)/(-1(-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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2\u03c0(4x^2 + 1x + 4) - 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) = (-2(2|(3sin(1\u03c0x - 2) + 1)| + 1))/(-1(2|(3sin(1\u03c0x - 2) + 1)| + 1) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1((3(-1|x + 3| + 3) - 2)/(2(-1|x + 3| + 3) + 2))^2 + 1((3(-1|x + 3| + 3) - 2)/(2(-1|x + 3| + 3) + 2)) + 5) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(4x^2 - 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": "47", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((-1(-3(-3|x + 1| + 3)^2 + 3(-3|x + 1| + 3) + 4) - 2)/(1(-3(-3|x + 1| + 3)^2 + 3(-3|x + 1| + 3) + 4) + 1)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(2sin(2\u03c0x - 2)))/(1(2sin(2\u03c0x - 2)) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1(3(-2sin(3\u03c0x + 3) + 1)^2 + 1(-2sin(3\u03c0x + 3) + 1) + 3) + 3)| + 3 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) = -3(-2|x - 1| + 1)^2 + 3(-2|x - 1| + 1) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3(1|(3cos(3\u03c0x - 3) - 2) - 3| + 2)^2 + 4(1|(3cos(3\u03c0x - 3) - 2) - 3| + 2) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1\u03c0(-3(-3|((2x + 3)/(1x - 3))| + 3)^2 - 2(-3|((2x + 3)/(1x - 3))| + 3)) + 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-3sin(2\u03c0x - 3)) + 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3((3(1cos(1\u03c0x + 2)) + 2)/(1(1cos(1\u03c0x + 2)) + 3))^2 + 4((3(1cos(1\u03c0x + 2)) + 2)/(1(1cos(1\u03c0x + 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) = 3|(5(-2x - 5)^2 - 5(-2x - 5) - 2) + 1| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2cos(1\u03c0(1(-3x^2 + 2x - 4) + 4) + 3)) + 3)/(2(2cos(1\u03c0(1(-3x^2 + 2x - 4) + 4) + 3)) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(3(1sin(3\u03c0x + 3) + 3)^2 - 3) - 3)/(-1(3(1sin(3\u03c0x + 3) + 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) = -1|(5x^2 + 3x + 1) - 1| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-3(-3|(-1sin(1\u03c0x - 2)) + 1| + 2) + 5)^2 - 5(-3(-3|(-1sin(1\u03c0x - 2)) + 1| + 2) + 5) - 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) = -2sin(2\u03c0(-3|x - 1| - 3) - 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-3sin(2\u03c0(2(2|x + 2| + 1) - 2) - 3))^2 - 4(-3sin(2\u03c0(2(2|x + 2| + 1) - 2) - 3)) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2\u03c0(1x + 5) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(1\u03c0((1(-5x - 1) + 3)/(-3(-5x - 1) + 2)) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(1cos(1\u03c0(5(-4x^2 + 2x - 3) - 1) - 1) + 3)| - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(1|(5(-5x^2 + 1x + 1) + 1)| - 2) + 2)/(1(1|(5(-5x^2 + 1x + 1) + 1)| - 2) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(1\u03c0(2(-3|((1x)/(2x + 2)) - 2| - 1)^2 - 3(-3|((1x)/(2x + 2)) - 2| - 1) + 4) - 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2|x + 1|)^2 + 1(2|x + 1|) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (1(2cos(1\u03c0x - 3)) + 3)/(1(2cos(1\u03c0x - 3)) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(2\u03c0(-3|x|) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((2(2sin(2\u03c0x - 2) - 2) - 1)/(-1(2sin(2\u03c0x - 2) - 2) - 2))^2 - 3((2(2sin(2\u03c0x - 2) - 2) - 1)/(-1(2sin(2\u03c0x - 2) - 2) - 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(-3cos(1\u03c0x + 3) + 3)^2 - 5(-3cos(1\u03c0x + 3) + 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) = 3((-1(2sin(3\u03c0(-1|x + 2| + 1)) + 2) - 2)/(3(2sin(3\u03c0(-1|x + 2| + 1)) + 2) + 2)) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2|x - 2| + 1)^2 - 3(-2|x - 2| + 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1cos(1\u03c0(3x^2 + 5x + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-1(-3|(-2cos(3\u03c0x + 3) - 1) + 2| - 2) - 3)/(1(-3|(-2cos(3\u03c0x + 3) - 1) + 2| - 2) + 1))^2 - 3((-1(-3|(-2cos(3\u03c0x + 3) - 1) + 2| - 2) - 3)/(1(-3|(-2cos(3\u03c0x + 3) - 1) + 2| - 2) + 1)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(1\u03c0((-2x - 1)/(2x - 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) = 2cos(1\u03c0(4x - 3) + 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(2|(3cos(2\u03c0(2x + 2) - 2) + 1) + 2|) + 1)/(-1(2|(3cos(2\u03c0(2x + 2) - 2) + 1) + 2|) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2e^(1((1(-2|(-4x^2 + 3x - 3) + 3| + 2) + 2)/(3(-2|(-4x^2 + 3x - 3) + 3| + 2) - 2))) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2sin(2\u03c0(-5(3((3x + 1)/(1x - 2)) + 2)^2 - 3(3((3x + 1)/(1x - 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": "35", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(1\u03c0(-1|(-4x + 5) + 3|) - 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(1(4((-1x + 3)/(3x + 2)) + 5)^2 + 5(4((-1x + 3)/(3x + 2)) + 5) + 3) + 2| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(-1x^2 + 4x + 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": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-3|(-3sin(1\u03c0x - 3) + 2) + 2| - 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1e^(1(-3(2|x + 1| - 2)^2 - 5(2|x + 1| - 2) + 4)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-1e^(-1(-5((-3x - 2)/(1x)) - 2)) + 1)^2 - 1(-1e^(-1(-5((-3x - 2)/(1x)) - 2)) + 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-2|(-2(-4x^2 - 4x + 4) + 3) - 2|) - 1)/(-3(-2|(-2(-4x^2 - 4x + 4) + 3) - 2|) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2(1sin(1\u03c0(-1(-1|x - 1|) + 4) - 3) - 1)^2 + 4(1sin(1\u03c0(-1(-1|x - 1|) + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(1\u03c0(-2|x - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(4(-4x^2 + 1x - 1) + 3) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1\u03c0(-2|x + 3| + 1) - 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) = (-2(-1|(3(4x^2 + 2x - 5) - 2) - 3| - 2))/(3(-1|(3(4x^2 + 2x - 5) - 2) - 3| - 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1sin(1\u03c0(-4x^2 + 1x - 1) + 1) - 2) + 2| + 2 is strictly increasing.\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|(-2x^2 - 5x + 2) + 3| + 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-1sin(2\u03c0(4x + 3) - 1) + 2) - 2| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(1cos(2\u03c0(-5(1|x + 3| + 3) + 3) + 3) - 2) - 3)/(3(1cos(2\u03c0(-5(1|x + 3| + 3) + 3) + 3) - 2) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3|(-1(3sin(1\u03c0((1x - 1)/(-1x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2cos(2\u03c0x - 2) + 2)^2 - 5(-2cos(2\u03c0x - 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(-2sin(2\u03c0(-3((2x + 3)/(-3x - 2))^2 - 5((2x + 3)/(-3x - 2)) - 2) - 1) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-1sin(2\u03c0x - 1) - 3)^2 + 3(-1sin(2\u03c0x - 1) - 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-5(2cos(3\u03c0x - 3) + 3)^2 + 5(2cos(3\u03c0x - 3) + 3) - 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) = -2e^(1(-1|((-1x)/(-3x + 1)) - 3| - 3)) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-4(-1(1|x + 1| - 3) - 3)^2 - 4(-1(1|x + 1| - 3) - 3) + 3) - 1)/(1(-4(-1(1|x + 1| - 3) - 3)^2 - 4(-1(1|x + 1| - 3) - 3) + 3) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1sin(2\u03c0(-4x + 1)) + 3)^2 + 1(1sin(2\u03c0(-4x + 1)) + 3) + 3 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) = 2(-2|(-2cos(3\u03c0x + 3) - 3) - 2|)^2 - 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) = -4(-2sin(2\u03c0((1x)/(-3x + 2)) + 2)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-3(1((-1x - 3)/(2x - 1))^2 - 2((-1x - 3)/(2x - 1)) + 3) - 1)| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-2(-1|(1x^2 + 4x + 4) - 1| - 3) - 1)/(-3(-1|(1x^2 + 4x + 4) - 1| - 3) + 2)) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-2|(-5x^2 + 5x + 2)| + 1) - 2)/(2(-2|(-5x^2 + 5x + 2)| + 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(3|(-5x^2 + 5x + 1) + 2| - 1) + 1)/(-1(3|(-5x^2 + 5x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(-1((3(-1x^2 - 3x - 1) - 2)/(-1(-1x^2 - 3x - 1) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(4((-1(1x^2 - 5x) - 3)/(-2(1x^2 - 5x) - 3)) + 5) - 3| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(3\u03c0(-1|x - 2| + 3) + 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(5((-2x - 2)/(1x - 2))^2 - 1((-2x - 2)/(1x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3\u03c0(2((1(-2|x| - 1) + 3)/(3(-2|x| - 1) - 2)) + 4) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-2sin(1\u03c0x - 1) + 1))/(-2(-2sin(1\u03c0x - 1) + 1) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1sin(1\u03c0((3(3x + 5) + 1)/(-1(3x + 5) + 2)) + 2) + 2)^2 - 3(-1sin(1\u03c0((3(3x + 5) + 1)/(-1(3x + 5) + 2)) + 2) + 2) + 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(-3cos(3\u03c0x - 2)), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|(1sin(2\u03c0x) - 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2(5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) - 4) + 2) - 1)/(2(-2(5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) - 4) + 2) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(2((2x - 1)/(2x + 2)) - 4) + 1| + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-2(-3cos(3\u03c0x + 2) - 3) - 1)/(3(-3cos(3\u03c0x + 2) - 3) + 1))^2 + 1((-2(-3cos(3\u03c0x + 2) - 3) - 1)/(3(-3cos(3\u03c0x + 2) - 3) + 1)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1cos(3\u03c0x + 1) + 3) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((3(-3(-2sin(2\u03c0x - 3) + 3)^2 + 4(-2sin(2\u03c0x - 3) + 3) - 5) - 1)/(-2(-3(-2sin(2\u03c0x - 3) + 3)^2 + 4(-2sin(2\u03c0x - 3) + 3) - 5) + 3)) + 2| - 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) = 1cos(3\u03c0(-4x - 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": "79", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(-1sin(2\u03c0x + 2) - 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3(-3cos(2\u03c0x + 3) + 3)^2 - 4(-3cos(2\u03c0x + 3) + 3) + 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(3\u03c0(-5(-1|(-4x^2 + 3x + 4) + 3| - 2) + 5) + 3) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3|(1(-2cos(2\u03c0x - 1) + 3) - 2) - 2| + 1)^2 - 3(3|(1(-2cos(2\u03c0x - 1) + 3) - 2) - 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": "For the function f(x) = -3(-2sin(2\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2\u03c0(-2((-3x + 1)/(-1x - 1))^2 + 4((-3x + 1)/(-1x - 1)) - 1) + 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-2|((-2(2cos(2\u03c0x + 1) + 2) + 1)/(2(2cos(2\u03c0x + 1) + 2) + 2))| - 2)^2 + 2(-2|((-2(2cos(2\u03c0x + 1) + 2) + 1)/(2(2cos(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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2e^(-2(-2|x| + 1)) + 1)^2 + 3(-2e^(-2(-2|x| + 1)) + 1) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-3sin(1\u03c0(-1|x - 2| - 3) - 2) + 3)^2 + 5(-3sin(1\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": "11", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2\u03c0(-4x^2 - 4) + 2) positive?\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) = 1(2|(-2cos(1\u03c0x + 2) - 1) + 3| + 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(2((2(-3|x - 3|) + 3)/(1(-3|x - 3|) - 2))^2 - 5((2(-3|x - 3|) + 3)/(1(-3|x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(2cos(1\u03c0x + 1) - 1) + 2)/(2(2cos(1\u03c0x + 1) - 1) + 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) = (-2(2cos(2\u03c0x + 2) - 3))/(-3(2cos(2\u03c0x + 2) - 3) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(3(3x^2 + 3x - 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": "2", "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) - 1| + 3) + 3)/(-2(1|(-1sin(3\u03c0x - 3) + 2) - 1| + 3) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((2(-3(-3|x + 3|) - 5) + 3)/(-1(-3(-3|x + 3|) - 5)))^2 - 2((2(-3(-3|x + 3|) - 5) + 3)/(-1(-3(-3|x + 3|) - 5))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2|(-5x + 2) + 3| + 3)^2 - 3(-2|(-5x + 2) + 3| + 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(1\u03c0((-1x + 3)/(1x)) + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(4(3|(5x^2 + 4x - 5) + 3| + 3) + 1) + 3)/(3(4(3|(5x^2 + 4x - 5) + 3| + 3) + 1) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(3|(-3((2x + 3)/(-3x + 2)) - 2) - 1| - 1)^2 + 1(3|(-3((2x + 3)/(-3x + 2)) - 2) - 1| - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1|(1cos(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": "10", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-5(5((-2x - 2)/(1x + 2)))^2 + 4(5((-2x - 2)/(1x + 2))) + 3) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(-2((2x + 2)/(2x - 1))^2 + 2((2x + 2)/(2x - 1)) - 2) + 1| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (3(-2|(2(-1cos(3\u03c0x + 1) + 3)^2 + 1(-1cos(3\u03c0x + 1) + 3))| + 1) + 1)/(3(-2|(2(-1cos(3\u03c0x + 1) + 3)^2 + 1(-1cos(3\u03c0x + 1) + 3))| + 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3e^(1(3((-1x)/(2x - 1))^2 + 2)) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(4((1(2|x - 3| + 1) + 3)/(3(2|x - 3| + 1) - 2)) - 3)^2 - 4(4((1(2|x - 3| + 1) + 3)/(3(2|x - 3| + 1) - 2)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-1|(2sin(2\u03c0x - 1) + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -5(1cos(1\u03c0x + 2) + 2)^2 - 5(1cos(1\u03c0x + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2cos(1\u03c0(2(-1|((-3x - 3)/(3x - 1)) + 1| + 2)^2 - 3(-1|((-3x - 3)/(3x - 1)) + 1| + 2) - 5) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-1(3|((3x)/(-1x - 2)) + 2| - 2)^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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (-1(-1cos(1\u03c0(-2(3|x - 3| - 2)^2 + 5(3|x - 3| - 2) + 2) + 3)) - 1)/(1(-1cos(1\u03c0(-2(3|x - 3| - 2)^2 + 5(3|x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((2(-2|x - 3| + 2) - 1)/(-3(-2|x - 3| + 2) - 2))^2 + 4((2(-2|x - 3| + 2) - 1)/(-3(-2|x - 3| + 2) - 2)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1(-3|x - 2| + 1)^2 + 1(-3|x - 2| + 1) - 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((2(2cos(3\u03c0(-4x + 3) + 2) - 2) + 1)/(-1(2cos(3\u03c0(-4x + 3) + 2) - 2) + 3))^2 + 2((2(2cos(3\u03c0(-4x + 3) + 2) - 2) + 1)/(-1(2cos(3\u03c0(-4x + 3) + 2) - 2) + 3)) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2sin(2\u03c0(-5(-1|(-1x^2 - 1x - 3) - 1| - 2) - 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(2|(2cos(2\u03c0x - 2) - 2) - 1| - 2)^2 - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-3e^(-1(-2((-1x - 2)/(3x))^2 - 1((-1x - 2)/(3x)) - 5)) - 2) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3cos(3\u03c0(-3|((1x + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2\u03c0(-2|x - 2| + 2)) + 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(4((-2(-2|x + 2| + 3) + 2)/(-2(-2|x + 2| + 3) + 1))^2 - 4) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2cos(2\u03c0x + 3) + 1) + 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) = -3cos(1\u03c0(1(5x^2 + 2x - 5) - 4) - 2) is strictly increasing.\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(2cos(3\u03c0x + 3) - 3))/(2(2cos(3\u03c0x + 3) - 3) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(1sin(2\u03c0((1x)/(3x - 2)) + 3)) - 1| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((-2x)/(1x + 2)) - 1| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1|((-3(1x + 4) + 2)/(1(1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(3sin(2\u03c0(2x - 4) - 2) + 2) - 1)/(3(3sin(2\u03c0(2x - 4) - 2) + 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) = 2cos(2\u03c0(-5x + 2) + 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|((-2(1x + 1) + 3)/(-3(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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(5((1x + 1)/(3x - 2)) - 1)| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1((-3(1sin(3\u03c0x + 1) + 2) + 1)/(-3(1sin(3\u03c0x + 1) + 2) + 2))^2 + 5((-3(1sin(3\u03c0x + 1) + 2) + 1)/(-3(1sin(3\u03c0x + 1) + 2) + 2)) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((-2(1x^2 + 3x - 2))/(-1(1x^2 + 3x - 2) - 1)) + 2| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3\u03c0(1((-3(4x^2 + 1x - 2))/(2(4x^2 + 1x - 2) + 1)) + 1) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1sin(2\u03c0(4x - 4) - 3) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0((2x - 2)/(-1x - 2)) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4((1(-3(-3|x - 3| + 2) + 3) - 1)/(-2(-3(-3|x - 3| + 2) + 3) + 1))^2 - 4((1(-3(-3|x - 3| + 2) + 3) - 1)/(-2(-3(-3|x - 3| + 2) + 3) + 1)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3|(-1e^(-2(-2(-3x^2 + 1x) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2((2(1x) + 1)/(1(1x) - 2))^2 - 1((2(1x) + 1)/(1(1x) - 2)) + 2) + 3| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(3(-2sin(2\u03c0(-1|x| - 2) - 3) + 1)^2 - 2(-2sin(2\u03c0(-1|x| - 2) - 3) + 1) + 3) - 5 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) = -5(1|(1x^2 + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((-1(5(-1x - 1)^2 + 2(-1x - 1) - 4) - 2)/(-3(5(-1x - 1)^2 + 2(-1x - 1) - 4) + 2)) + 1| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(3cos(1\u03c0(-1((-2x - 1)/(-3x + 1)) + 5) + 2) - 2)^2 + 4(3cos(1\u03c0(-1((-2x - 1)/(-3x + 1)) + 5) + 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = (2(-1|(-2sin(2\u03c0x - 1) + 3)| - 2))/(-1(-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": "20", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -2|(3(-1x^2 - 5x + 3) - 4) + 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((1(5(1cos(2\u03c0x - 2) + 2)) + 1)/(-3(5(1cos(2\u03c0x - 2) + 2)) - 1))^2 + 1((1(5(1cos(2\u03c0x - 2) + 2)) + 1)/(-3(5(1cos(2\u03c0x - 2) + 2)) - 1)) - 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) = -3cos(1\u03c0(-1|x - 1| - 2) + 1) - 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) = (-2(-1sin(3\u03c0x + 2) - 1) - 2)/(-3(-1sin(3\u03c0x + 2) - 1) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(1x^2 - 2x - 1) - 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3\u03c0((-1(1x^2 + 2x - 3) + 3)/(-1(1x^2 + 2x - 3))) + 2) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((2(-2(1x^2 - 5x - 3) - 3) + 1)/(-2(-2(1x^2 - 5x - 3) - 3) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1|((-3(2cos(3\u03c0x + 2) - 1) + 3)/(-3(2cos(3\u03c0x + 2) - 1) - 3)) + 2|)^2 + 1(-1|((-3(2cos(3\u03c0x + 2) - 1) + 3)/(-3(2cos(3\u03c0x + 2) - 1) - 3)) + 2|) - 3 is strictly increasing.\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(-2sin(3\u03c0(3(-3x + 5)^2 - 3(-3x + 5)) + 3) - 3) + 2)/(-2(-2sin(3\u03c0(3(-3x + 5)^2 - 3(-3x + 5)) + 3) - 3) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1sin(1\u03c0(-3((1x - 1)/(-2x + 1)) - 1) - 2) - 2)^2 + 2(1sin(1\u03c0(-3((1x - 1)/(-2x + 1)) - 1) - 2) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-1sin(3\u03c0((3(1x^2 - 5x + 2))/(3(1x^2 - 5x + 2) + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(4((-2(-2sin(1\u03c0x + 1)) - 2)/(-1(-2sin(1\u03c0x + 1)))) + 3)^2 + 3(4((-2(-2sin(1\u03c0x + 1)) - 2)/(-1(-2sin(1\u03c0x + 1)))) + 3) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3\u03c0(-2|(4x + 1)| - 3) + 1) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((1(-3(2x - 4)^2 - 4(2x - 4) + 2) + 3)/(2(-3(2x - 4)^2 - 4(2x - 4) + 2) + 3)) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1cos(1\u03c0(2(-1x^2 - 5) + 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": "51", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((-2(-1(2x^2 - 5x + 1) + 5) + 2)/(-1(-1(2x^2 - 5x + 1) + 5) - 3)) + 3| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(2sin(1\u03c0(3|x - 1| + 1) + 3) + 1)^2 - 3(2sin(1\u03c0(3|x - 1| + 1) + 3) + 1) + 4 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) = 1e^(2(-4(5x^2 + 1x - 5) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1(-3(2sin(3\u03c0x + 2) - 1) - 4)^2 - 2(-3(2sin(3\u03c0x + 2) - 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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(5((3x - 3)/(-3x + 2)) + 3) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3sin(3\u03c0((-3x - 1)/(2x + 2)) - 1) - 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) = -1(4(-1|x - 1|) + 5)^2 - 5(4(-1|x - 1|) + 5) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 4(-2|(-2cos(3\u03c0((1x - 2)/(-1x + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(1|(3((-2x - 3)/(3x - 2)) + 5) + 1| - 3)^2 - 1(1|(3((-2x - 3)/(3x - 2)) + 5) + 1| - 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((3(2|x| - 2) + 3)/(1(2|x| - 2)))^2 + 5((3(2|x| - 2) + 3)/(1(2|x| - 2))) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1|((-1(3e^(2x) - 3) - 2)/(-1(3e^(2x) - 3) + 2)) + 3| - 2)^2 + 5(1|((-1(3e^(2x) - 3) - 2)/(-1(3e^(2x) - 3) + 2)) + 3| - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -1sin(2\u03c0((2x - 3)/(-2x - 2))) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3(1cos(2\u03c0((-1x - 2)/(-1x - 3)) - 2) - 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1cos(2\u03c0(-2|x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(1\u03c0(-4(2|x - 2| + 3)^2 - 5(2|x - 2| + 3) - 5) + 1) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -4(3|((-2(2sin(2\u03c0x + 3) + 3) - 1)/(1(2sin(2\u03c0x + 3) + 3))) - 3| - 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((3(1|x| - 2) - 3)/(2(1|x| - 2) + 1))^2 - 3((3(1|x| - 2) - 3)/(2(1|x| - 2) + 1)) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3cos(3\u03c0((-2x - 1)/(-3x - 2)) + 1) - 2)^2 - 1(3cos(3\u03c0((-2x - 1)/(-3x - 2)) + 1) - 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 3sin(1\u03c0(-5((-3(4x - 1) - 2)/(-3(4x - 1)))^2) - 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2(4(3cos(3\u03c0x - 2) - 1) - 4)^2 + 4(4(3cos(3\u03c0x - 2) - 1) - 4) + 4) + 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((-1x + 3)/(-1x + 2))^2 + 5((-1x + 3)/(-1x + 2)) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-3sin(1\u03c0x - 3)) + 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((2(-3|(-3x^2 - 4x + 3) - 2| - 1) - 3)/(2(-3|(-3x^2 - 4x + 3) - 2| - 1) + 1)) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1e^(-2(2(-4x)^2 + 4(-4x) + 2)) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 1(-1e^(-1((-2x - 3)/(-2x - 1))) - 1)^2 + 3(-1e^(-1((-2x - 3)/(-2x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = 2|(-1((-2x + 2)/(1x + 2)) + 3) - 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "For the function f(x) = -3sin(2\u03c0((3(1(2|x - 1| - 3)^2 - 1(2|x - 1| - 3) + 4))/(3(1(2|x - 1| - 3)^2 - 1(2|x - 1| - 3) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2\u03c0(-4(-3x - 4)^2 + 1(-3x - 4) - 4) + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(4(4((1x)/(-2x + 1)) + 1)^2 + 2) + 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2cos(1\u03c0x + 3) - 2)^2 - 2(2cos(1\u03c0x + 3) - 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}