{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2|(3sin(2\u03c0(5x^2 + 5x + 1) + 3) + 1) + 3| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "502", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (3(-1sin(1\u03c0(1x^2 + 3x - 2) + 2) + 3) - 1)/(2(-1sin(1\u03c0(1x^2 + 3x - 2) + 2) + 3) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(1|(-2sin(2\u03c0((1x - 2)/(-3x - 3)) + 2) - 1) + 1| - 2)^2 - 2(1|(-2sin(2\u03c0((1x - 2)/(-3x - 3)) + 2) - 1) + 1| - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(1\u03c0(3(1|(-2x - 3) + 2| + 1)^2 + 4(1|(-2x - 3) + 2| + 1) + 1) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "267", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(1(1cos(1\u03c0x + 1) + 1) + 3) + 2)/(2(1(1cos(1\u03c0x + 1) + 1) + 3) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(1\u03c0(2|(3x - 2) + 2| + 2) - 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2((-3(-2cos(1\u03c0x + 3)) - 1)/(-2(-2cos(1\u03c0x + 3)) + 3)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(1sin(1\u03c0(3x^2 - 1x - 2) + 3) + 1) + 3)/(-1(1sin(1\u03c0(3x^2 - 1x - 2) + 3) + 1) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "567", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2|(2sin(1\u03c0(-1x + 3) + 1) - 3)| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3(1sin(2\u03c0(-5(-2|x + 1| - 1) - 1) - 3) + 3) + 3)/(-2(1sin(2\u03c0(-5(-2|x + 1| - 1) - 1) - 3) + 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "194", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2|(-4(1cos(1\u03c0x)) - 5) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(1|(-1sin(1\u03c0x) + 2) + 2| - 3)^2 - 1(1|(-1sin(1\u03c0x) + 2) + 2| - 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3|(-1cos(1\u03c0(1(-3x + 5)^2 + 1(-3x + 5) - 5) - 3) - 1) + 1| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "494", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4((-1(-3cos(3\u03c0(-3x^2 - 4)) + 3) - 2)/(3(-3cos(3\u03c0(-3x^2 - 4)) + 3) - 2)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "537", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|(1(-1cos(1\u03c0((2x - 1)/(1x)) + 1) - 2)^2 - 3(-1cos(1\u03c0((2x - 1)/(1x)) + 1) - 2) + 4) + 2| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3cos(2\u03c0(1|((2x)/(-2x - 1)) + 2|) + 2) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1(-2e^(1(4x - 1)) + 1)^2 + 2(-2e^(1(4x - 1)) + 1) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(2cos(1\u03c0(2|x|) + 3)) + 1)/(-1(2cos(1\u03c0(2|x|) + 3)) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3sin(1\u03c0((-3(1x + 2) + 3)/(-2(1x + 2))) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-3(1(2e^(-2x) - 3)^2 + 1(2e^(-2x) - 3) - 2) - 2)/(-3(1(2e^(-2x) - 3)^2 + 1(2e^(-2x) - 3) - 2) + 4) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2cos(3\u03c0(-2|((1x)/(2x + 1)) - 1| - 1) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2cos(2\u03c0(5(-2|(-1x^2 - 5x + 1) + 1| + 2) + 3) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "244", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1cos(3\u03c0(-3(-4(1|x - 3| + 1)^2 - 1(1|x - 3| + 1) - 5) + 2) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "243", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|(-3sin(2\u03c0(-4((-2x + 2)/(-1x + 2))^2 + 3((-2x + 2)/(-1x + 2))) - 1) + 1) - 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "153", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3(1cos(2\u03c0(1|x + 3| + 2) + 2) + 2)^2 - 2(1cos(2\u03c0(1|x + 3| + 2) + 2) + 2) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2cos(1\u03c0(4(-1((-3x - 3)/(-2x))^2 + 1((-3x - 3)/(-2x)) - 2) + 5) + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(1\u03c0(-3((-2x - 3)/(-3x - 3))^2 + 1((-2x - 3)/(-3x - 3))) - 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(1|(4(-2cos(1\u03c0x - 3)) + 5) + 2| + 2) + 2)/(1(1|(4(-2cos(1\u03c0x - 3)) + 5) + 2| + 2) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(-2cos(3\u03c0(4x^2 + 5x - 5) + 3) + 1) - 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "497", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(2cos(1\u03c0((1x - 1)/(1x - 2)) - 3))^2 - 2(2cos(1\u03c0((1x - 1)/(1x - 2)) - 3)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (2(-2sin(2\u03c0(-1x^2 - 1x + 2) + 2) - 1))/(2(-2sin(2\u03c0(-1x^2 - 1x + 2) + 2) - 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "236", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(2\u03c0(5((-3x - 1)/(-1x + 2))^2 - 5((-3x - 1)/(-1x + 2)) + 3) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "127", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(-2sin(1\u03c0(2x^2 - 3x - 5) + 3)) - 3)/(-1(-2sin(1\u03c0(2x^2 - 3x - 5) + 3)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "294", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1|(1(-2cos(3\u03c0(1x + 2) - 2) - 2)^2 + 3(-2cos(3\u03c0(1x + 2) - 2) - 2) - 2) + 1| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "173", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 5(-4(-1sin(2\u03c0((1x + 1)/(3x)) - 2) - 1)^2 + 4(-1sin(2\u03c0((1x + 1)/(3x)) - 2) - 1) - 5) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(3((2(1cos(2\u03c0x - 3) + 1) + 1)/(1(1cos(2\u03c0x - 3) + 1) - 3))^2 + 3((2(1cos(2\u03c0x - 3) + 1) + 1)/(1(1cos(2\u03c0x - 3) + 1) - 3)) + 3) - 1| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -5(-3sin(1\u03c0((2x - 1)/(-3x)) + 3) + 3)^2 - 4(-3sin(1\u03c0((2x - 1)/(-3x)) + 3) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(3cos(3\u03c0((-3x - 1)/(-3x)) + 3) + 1)^2 - 3(3cos(3\u03c0((-3x - 1)/(-3x)) + 3) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3sin(1\u03c0(5(-3((-1x + 1)/(-2x - 3)) + 2)^2 - 3(-3((-1x + 1)/(-2x - 3)) + 2) - 4)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3cos(3\u03c0((-1(2(2|x + 2| - 2)^2 + 1(2|x + 2| - 2) - 4) - 3)/(-2(2(2|x + 2| - 2)^2 + 1(2|x + 2| - 2) - 4) + 3)) + 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3cos(3\u03c0((1(-3|x + 2| + 3) - 2)/(-1(-3|x + 2| + 3) - 2))) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3sin(2\u03c0((1(-1x - 3) + 2)/(2(-1x - 3) - 1)) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(3\u03c0(2|(3x - 2) - 1| - 2) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(-2cos(2\u03c0(2|(2x^2 - 4x + 2) - 3| + 1) + 2) - 2) + 2)/(-3(-2cos(2\u03c0(2|(2x^2 - 4x + 2) - 3| + 1) + 2) - 2) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "440", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1cos(3\u03c0((-3(-1x^2 + 3x - 3) - 3)/(3(-1x^2 + 3x - 3) + 1)) - 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|((1(3sin(1\u03c0(2x^2 - 4) - 3) + 3) - 1)/(-1(3sin(1\u03c0(2x^2 - 4) - 3) + 3) - 1)) + 1| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "401", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|(-1((-2(3sin(1\u03c0x + 3) + 1) + 3)/(-2(3sin(1\u03c0x + 3) + 1)))^2 - 1((-2(3sin(1\u03c0x + 3) + 1) + 3)/(-2(3sin(1\u03c0x + 3) + 1))) - 5) + 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(1|(3e^(-1(4x + 5))) - 1| - 2)^2 - 3(1|(3e^(-1(4x + 5))) - 1| - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2sin(2\u03c0(-1|(5x^2 - 5x + 1) - 3| + 3) + 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "507", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(-3|(4(-1sin(1\u03c0x + 1) + 2) + 2) - 1| - 2)^2 - 5(-3|(4(-1sin(1\u03c0x + 1) + 2) + 2) - 1| - 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-2(-2|(2e^(-2(1x - 5))) + 1| - 3))/(3(-2|(2e^(-2(1x - 5))) + 1| - 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (1(1|(2cos(3\u03c0(-2x + 5) - 1) + 2) - 3| - 3))/(3(1|(2cos(3\u03c0(-2x + 5) - 1) + 2) - 3| - 3) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4((3(1cos(1\u03c0(1|x - 3| - 1) + 2) + 2) + 2)/(3(1cos(1\u03c0(1|x - 3| - 1) + 2) + 2) + 1))^2 + 3((3(1cos(1\u03c0(1|x - 3| - 1) + 2) + 2) + 2)/(3(1cos(1\u03c0(1|x - 3| - 1) + 2) + 2) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-3(1(2sin(1\u03c0(1|x - 2| + 1)))) + 1)/(3(1(2sin(1\u03c0(1|x - 2| + 1))))) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1sin(3\u03c0(-4(-3(-3|x - 3| + 1))^2 + 2(-3(-3|x - 3| + 1)) - 5) + 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-1|(-2sin(1\u03c0(1x - 4) - 1) - 1) - 1| + 3) + 3)/(-1(-1|(-2sin(1\u03c0(1x - 4) - 1) - 1) - 1| + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(-3|(2sin(2\u03c0x - 2)) + 1| - 3))/(3(-3|(2sin(2\u03c0x - 2)) + 1| - 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-1cos(2\u03c0(2(1|x + 1| + 3) - 2) - 3) + 3) + 2)/(-2(-1cos(2\u03c0(2(1|x + 1| + 3) - 2) - 3) + 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1cos(2\u03c0(-3|(1(-1x - 4)^2 - 3(-1x - 4) + 5) + 2|)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "264", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(2cos(1\u03c0(5x + 2) - 2) + 3) - 1)/(1(2cos(1\u03c0(5x + 2) - 2) + 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-3cos(3\u03c0(-5x^2 - 3x - 4) + 3)))/(3(-3cos(3\u03c0(-5x^2 - 3x - 4) + 3)) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1cos(2\u03c0(3((-2x)/(-3x + 1)) + 3) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2(-3|(-1sin(2\u03c0((-1x - 2)/(1x)))) - 2| - 1)^2 - 2(-3|(-1sin(2\u03c0((-1x - 2)/(1x)))) - 2| - 1) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3|(-2(-3e^(1(-1x^2 + 5x + 1)) - 3) + 5) + 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(4(2sin(1\u03c0x + 3) - 2) - 5) + 2)/(-3(4(2sin(1\u03c0x + 3) - 2) - 5) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(-3sin(1\u03c0((-1(-3|x - 2| + 3) - 3)/(-2(-3|x - 2| + 3) + 3)) + 2) + 3)^2 + 2(-3sin(1\u03c0((-1(-3|x - 2| + 3) - 3)/(-2(-3|x - 2| + 3) + 3)) + 2) + 3) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3cos(1\u03c0(4(-2|(1x^2 + 3x + 3) + 3| + 3) - 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "251", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(3\u03c0((2(2|(1x^2 + 3x) - 3| + 1) + 1)/(-1(2|(1x^2 + 3x) - 3| + 1) - 2)) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3|(-5(1(1sin(1\u03c0x - 1) - 3) + 5)^2 - 3(1(1sin(1\u03c0x - 1) - 3) + 5) - 4) + 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3(3|(-3sin(1\u03c0x + 1) + 3) - 2| + 3)^2 - 2(3|(-3sin(1\u03c0x + 1) + 3) - 2| + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2sin(1\u03c0(-5(1|x + 1| - 2) + 5) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2((-2(-2sin(2\u03c0x) - 1) - 1)/(2(-2sin(2\u03c0x) - 1) + 3)) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3|(3sin(2\u03c0(1x - 2) + 1) + 3) + 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(3|(5((-1x)/(3x)) + 1) + 3| + 1)^2 + 4(3|(5((-1x)/(3x)) + 1) + 3| + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(-2sin(3\u03c0(-1|(5x^2 + 2x - 5) - 3| - 1) + 3) - 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "256", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1(2((3(-2cos(2\u03c0x - 3) + 2) - 1)/(-2(-2cos(2\u03c0x - 3) + 2) - 3)) - 2)^2 + 5(2((3(-2cos(2\u03c0x - 3) + 2) - 1)/(-2(-2cos(2\u03c0x - 3) + 2) - 3)) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1cos(1\u03c0(-5((-2(-1|x - 2| - 1) - 3)/(-1(-1|x - 2| - 1))) - 3) + 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1cos(1\u03c0(-2|(-3x - 2)| - 3) - 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(1\u03c0(2((3x + 2)/(-3x - 1)) - 2) - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(-1(3(-3sin(2\u03c0x + 2) - 1)^2 + 4(-3sin(2\u03c0x + 2) - 1) + 3) - 3) - 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (2(1(-3cos(2\u03c0x - 3) + 1)^2 - 5(-3cos(2\u03c0x - 3) + 1) + 5) + 3)/(1(1(-3cos(2\u03c0x - 3) + 1)^2 - 5(-3cos(2\u03c0x - 3) + 1) + 5) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-1(-5(-1|(1cos(3\u03c0x + 2)) + 2| + 1)^2 + 3(-1|(1cos(3\u03c0x + 2)) + 2| + 1) - 3) - 1)/(-3(-5(-1|(1cos(3\u03c0x + 2)) + 2| + 1)^2 + 3(-1|(1cos(3\u03c0x + 2)) + 2| + 1) - 3) + 4) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(-4(2cos(2\u03c0(3|x - 1| - 2) + 3) - 1)^2 - 5(2cos(2\u03c0(3|x - 1| - 2) + 3) - 1)) - 1)/(-2(-4(2cos(2\u03c0(3|x - 1| - 2) + 3) - 1)^2 - 5(2cos(2\u03c0(3|x - 1| - 2) + 3) - 1)) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1(1|((-2(-2x - 5))/(-3(-2x - 5)))| + 1)^2 - 4(1|((-2(-2x - 5))/(-3(-2x - 5)))| + 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2((-1(3cos(1\u03c0(1|x - 1| + 1) - 1) + 3) + 1)/(2(3cos(1\u03c0(1|x - 1| + 1) - 1) + 3) - 1)) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2cos(1\u03c0(-1((-1x + 1)/(3x))^2 + 5((-1x + 1)/(3x)) + 1)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-2sin(2\u03c0(1x^2 - 1x - 1) - 2) + 1))/(3(-2sin(2\u03c0(1x^2 - 1x - 1) - 2) + 1) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|((2(-1e^(-1(-5x - 3)) - 3) - 2)/(3(-1e^(-1(-5x - 3)) - 3))) - 1| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2|(1((-1(1sin(2\u03c0x + 1) + 1))/(2(1sin(2\u03c0x + 1) + 1) - 1)) + 2) + 2| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(1\u03c0(1|((-2(-2x^2 + 5x - 1))/(-3(-2x^2 + 5x - 1) + 1)) - 3| - 2)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1((3(-4(1sin(2\u03c0x - 1) + 3) - 4) - 1)/(-2(-4(1sin(2\u03c0x - 1) + 3) - 4) - 3))^2 - 2((3(-4(1sin(2\u03c0x - 1) + 3) - 4) - 1)/(-2(-4(1sin(2\u03c0x - 1) + 3) - 4) - 3)) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3(-1cos(3\u03c0(1x - 3) - 3))^2 + 4(-1cos(3\u03c0(1x - 3) - 3)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2|(5(-2sin(3\u03c0(1x + 4) + 1) - 3)^2 + 5(-2sin(3\u03c0(1x + 4) + 1) - 3) - 5) - 1| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(1\u03c0((-1(3|(3x^2 + 1x - 3) + 2| - 1) - 2)/(1(3|(3x^2 + 1x - 3) + 2| - 1))) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2sin(1\u03c0(1|((2x - 1)/(-2x - 1)) - 2| - 2) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-3(-3(2cos(2\u03c0(1|x| + 3) - 3) - 2)) + 3)/(-1(-3(2cos(2\u03c0(1|x| + 3) - 3) - 2)) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1sin(1\u03c0(1(2|((-3x + 1)/(-1x)) + 3| - 1)) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(-1|(-3cos(3\u03c0x) + 1) + 2| - 2)^2 + 4(-1|(-3cos(3\u03c0x) + 1) + 2| - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5((-3(1sin(3\u03c0x + 3) + 2) + 1)/(-2(1sin(3\u03c0x + 3) + 2) - 1)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(2\u03c0(-3|(-1x - 5) + 1| - 3)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4((2(3sin(1\u03c0x - 1) - 3) + 2)/(1(3sin(1\u03c0x - 1) - 3) - 2))^2 + 4((2(3sin(1\u03c0x - 1) - 3) + 2)/(1(3sin(1\u03c0x - 1) - 3) - 2)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1|(2sin(2\u03c0((-3(4x + 5) - 2)/(-2(4x + 5) - 1))) + 1) - 1| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(-1cos(2\u03c0((-1x - 3)/(-2x + 1)) - 2) + 3) - 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3cos(2\u03c0(-1((-2(3x^2 - 5x + 1) - 2)/(-3(3x^2 - 5x + 1) + 4))) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|(3(-2sin(3\u03c0x + 3) + 3) + 2) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1(-1sin(3\u03c0(5(3|x - 2| - 2)^2 - 3(3|x - 2| - 2) + 1)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1(-1(2sin(1\u03c0x - 3) - 2) + 3)^2 - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (3(3|(-3sin(2\u03c0x + 3) - 3) - 3| - 2) - 2)/(-3(3|(-3sin(2\u03c0x + 3) - 3) - 3| - 2) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3((1(-2sin(1\u03c0x - 1) - 1) + 2)/(1(-2sin(1\u03c0x - 1) - 1)))^2 - 5((1(-2sin(1\u03c0x - 1) - 1) + 2)/(1(-2sin(1\u03c0x - 1) - 1))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(1cos(3\u03c0(-2|(3x - 4) + 2|) - 3) + 1)^2 + 1(1cos(3\u03c0(-2|(3x - 4) + 2|) - 3) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "179", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(4(1cos(3\u03c0x + 3) + 1) - 3) + 1)/(-1(4(1cos(3\u03c0x + 3) + 1) - 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|(-1(3sin(2\u03c0((-1x + 1)/(-2x + 3))) - 2) + 5) - 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(3cos(2\u03c0(4x^2 + 2x - 2) + 3) + 2) - 2)/(-3(3cos(2\u03c0(4x^2 + 2x - 2) + 3) + 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "519", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2(-2sin(2\u03c0((-1(2|x - 3| + 3))/(-1(2|x - 3| + 3) + 2))) + 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|(2((2(2sin(1\u03c0x - 2) - 3))/(3(2sin(1\u03c0x - 2) - 3) + 1)) - 1) - 2| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(-2sin(3\u03c0(-5x + 2) + 3) - 2) - 1)/(-2(-2sin(3\u03c0(-5x + 2) + 3) - 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1sin(1\u03c0(-1|(-5((-2x + 1)/(3x - 2))^2 + 5)| - 2) + 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 4(-2sin(1\u03c0((-3(-1|x| - 3) + 1)/(1(-1|x| - 3) + 3)) + 1)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3cos(3\u03c0(-1|((-1(-5x^2 + 2x - 2) - 2)/(-1(-5x^2 + 2x - 2) - 3)) + 3| - 1)) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0(-5((2x - 2)/(-3x - 2))^2 + 1((2x - 2)/(-3x - 2)) - 5) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2sin(2\u03c0(-2(-1((-2x + 1)/(1x + 2)) + 3)^2 - 4(-1((-2x + 1)/(1x + 2)) + 3) + 1) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "140", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3((-2(1cos(2\u03c0(5x - 4) - 1) + 3) + 2)/(2(1cos(2\u03c0(5x - 4) - 1) + 3) - 1))^2 - 5((-2(1cos(2\u03c0(5x - 4) - 1) + 3) + 2)/(2(1cos(2\u03c0(5x - 4) - 1) + 3) - 1)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3((-1(1|(1cos(1\u03c0x - 3)) + 1| - 3) - 1)/(2(1|(1cos(1\u03c0x - 3)) + 1| - 3) - 1)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2cos(3\u03c0((1(2x^2 - 5x - 1) + 2)/(-2(2x^2 - 5x - 1) - 1)) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (2(-4(-1sin(2\u03c0(2|x + 2| + 2) - 1)) + 4))/(3(-4(-1sin(2\u03c0(2|x + 2| + 2) - 1)) + 4) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (3(-4(1|x + 2| + 2) + 2))/(-2(-4(1|x + 2| + 2) + 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2\u03c0((2(-1|(4x + 2) - 1| - 2) + 3)/(-1(-1|(4x + 2) - 1| - 2) + 3)) + 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2(1|(1sin(2\u03c0(-1x^2 + 3) + 3) - 1) + 3|) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(-3(2cos(1\u03c0x + 1) - 2) - 5) + 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1(-2|(3cos(3\u03c0(-2x - 4) - 3) + 2) - 1| - 3)^2 - 5(-2|(3cos(3\u03c0(-2x - 4) - 3) + 2) - 1| - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "239", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(2sin(3\u03c0(-2|x + 3| - 1) - 1) + 2)^2 - 3(2sin(3\u03c0(-2|x + 3| - 1) - 1) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2sin(1\u03c0(-2|(1x^2 + 4x - 2) - 2| - 2) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "209", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4(-3|((-2(2sin(2\u03c0x + 1) - 2) + 2)/(2(2sin(2\u03c0x + 1) - 2) + 1)) + 1|)^2 + 4(-3|((-2(2sin(2\u03c0x + 1) - 2) + 2)/(2(2sin(2\u03c0x + 1) - 2) + 1)) + 1|) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(3|(-2cos(2\u03c0((2x - 3)/(-1x + 2)) - 2) + 2)| - 3)^2 - 4(3|(-2cos(2\u03c0((2x - 3)/(-1x + 2)) - 2) + 2)| - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1sin(1\u03c0(1|(-5x + 2) - 1|) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1(3((-2(-2sin(1\u03c0x + 3)) - 3)/(2(-2sin(1\u03c0x + 3)) + 1))^2 + 2((-2(-2sin(1\u03c0x + 3)) - 3)/(2(-2sin(1\u03c0x + 3)) + 1)) + 5) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(2\u03c0(-2|(-4(3x^2 + 1x + 2) + 1) + 2| - 1) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "492", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(3((-2(1cos(1\u03c0x) + 2) - 3)/(3(1cos(1\u03c0x) + 2) + 3)) - 4)^2 - 5(3((-2(1cos(1\u03c0x) + 2) - 3)/(3(1cos(1\u03c0x) + 2) + 3)) - 4) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(-2cos(3\u03c0(-3x - 3) + 2) - 2) + 1)/(3(-2cos(3\u03c0(-3x - 3) + 2) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3|(-3cos(1\u03c0(-4x + 5) + 3) + 2) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2((2(-1e^(2(2|x + 1| - 1)) - 3) + 3)/(1(-1e^(2(2|x + 1| - 1)) - 3) + 1)) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1(3cos(1\u03c0(2|(-1x) + 3| - 2)) - 1)^2 - 2(3cos(1\u03c0(2|(-1x) + 3| - 2)) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-2|((-1(2sin(3\u03c0x) + 1) - 3)/(-3(2sin(3\u03c0x) + 1) - 3)) + 1|) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3(-3sin(1\u03c0(-3|x - 1| + 2)) + 3)^2 + 4(-3sin(1\u03c0(-3|x - 1| + 2)) + 3) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -4(3(1sin(3\u03c0x + 2) + 1)^2 + 5(1sin(3\u03c0x + 2) + 1) - 4) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(-4(3(3cos(1\u03c0x - 1) + 3))^2 + 4(3(3cos(1\u03c0x - 1) + 3)) + 1) + 1| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|((-2(1cos(1\u03c0x + 2) + 2) + 1)/(3(1cos(1\u03c0x + 2) + 2) - 2)) + 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2(-1(3cos(3\u03c0(2|x + 2| - 3) - 3) - 1)^2 + 5(3cos(3\u03c0(2|x + 2| - 3) - 3) - 1) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1sin(1\u03c0(1(3|x| + 1) + 1) - 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5(3|(1cos(3\u03c0x + 3)) + 1| + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5(-1sin(1\u03c0(-3x^2 + 3x - 3) - 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "284", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3sin(1\u03c0(4((-2x + 1)/(-1x)) - 4)) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|((1(-1sin(3\u03c0(5x^2 + 3x + 5) + 3) - 3) + 1)/(1(-1sin(3\u03c0(5x^2 + 3x + 5) + 3) - 3) + 2)) - 3| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "456", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4((-3(1|(1cos(1\u03c0x + 2)) - 2| + 2) - 2)/(1(1|(1cos(1\u03c0x + 2)) - 2| + 2)))^2 - 2((-3(1|(1cos(1\u03c0x + 2)) - 2| + 2) - 2)/(1(1|(1cos(1\u03c0x + 2)) - 2| + 2))) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2|(-1cos(2\u03c0(-3(-5x + 2)^2 + 5(-5x + 2)) + 3) + 2)| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "199", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|((-2(2sin(1\u03c0(2x^2 + 4x + 5) + 1) - 2) + 3)/(2(2sin(1\u03c0(2x^2 + 4x + 5) + 1) - 2) + 1)) + 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "493", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2\u03c0(2((-2(-2|x - 2| - 1) + 1)/(1(-2|x - 2| - 1) - 3))^2 + 3((-2(-2|x - 2| - 1) + 1)/(1(-2|x - 2| - 1) - 3)) + 4) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(-2|(3sin(2\u03c0x + 2) + 2)| + 3) - 3)/(-1(-2|(3sin(2\u03c0x + 2) + 2)| + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 5(5(2sin(3\u03c0x - 2) + 2)^2 + 2(2sin(3\u03c0x - 2) + 2) + 5) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-3sin(3\u03c0(-1|x| - 1)) - 2) + 3)/(-1(-3sin(3\u03c0(-1|x| - 1)) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(2cos(2\u03c0(2|((2x + 1)/(-1x - 2)) + 1| + 2) + 3) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5((-2(-3sin(1\u03c0x + 1)) + 3)/(-3(-3sin(1\u03c0x + 1)) + 4))^2 - 1((-2(-3sin(1\u03c0x + 1)) + 3)/(-3(-3sin(1\u03c0x + 1)) + 4)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-3(-1(1|(-2x - 4) - 2| - 1)^2 + 1) - 3)/(2(-1(1|(-2x - 4) - 2| - 1)^2 + 1) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "166", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2((-1(2sin(2\u03c0x) - 2) - 1)/(-2(2sin(2\u03c0x) - 2) + 1)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(4(1cos(3\u03c0x - 3) - 1)) + 2)/(1(4(1cos(3\u03c0x - 3) - 1)) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(1sin(2\u03c0((-1x + 2)/(1x)) - 3) - 2) - 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(3\u03c0((1(-1x + 3))/(2(-1x + 3) - 1)) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(-1(2cos(2\u03c0x + 2) + 2)^2 + 1(2cos(2\u03c0x + 2) + 2) - 3) + 1)/(-1(-1(2cos(2\u03c0x + 2) + 2)^2 + 1(2cos(2\u03c0x + 2) + 2) - 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2|(-3(1sin(2\u03c0(4x^2 - 5x + 4) - 3) - 1) + 4) - 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "256", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1cos(3\u03c0(-5((-2x - 1)/(-2x + 3)) + 5) + 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 5((2(-1sin(3\u03c0x + 1) + 2) - 1)/(1(-1sin(3\u03c0x + 1) + 2) + 1))^2 + 3((2(-1sin(3\u03c0x + 1) + 2) - 1)/(1(-1sin(3\u03c0x + 1) + 2) + 1)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 4(-2(2sin(2\u03c0(-1|x - 2| + 3) - 3) + 3) + 2)^2 - 1(-2(2sin(2\u03c0(-1|x - 2| + 3) - 3) + 3) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3(-1sin(3\u03c0((-1x + 3)/(2x + 2)) - 2) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3(-2cos(2\u03c0((2(1|x - 2| - 2) + 2)/(1(1|x - 2| - 2))) - 3) + 1)^2 - 5(-2cos(2\u03c0((2(1|x - 2| - 2) + 2)/(1(1|x - 2| - 2))) - 3) + 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "111", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3|(-4(4(-1cos(3\u03c0x + 2)) - 5)^2 - 1(4(-1cos(3\u03c0x + 2)) - 5) - 1) - 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-3(3(2cos(1\u03c0(4x + 2) - 1) - 2)^2 + 4(2cos(1\u03c0(4x + 2) - 1) - 2) - 1) + 2)/(-3(3(2cos(1\u03c0(4x + 2) - 1) - 2)^2 + 4(2cos(1\u03c0(4x + 2) - 1) - 2) - 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(1\u03c0((3(-3x + 5) - 2)/(-1(-3x + 5) - 3)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(3(-3|((-3x - 3)/(2x + 2)) + 3|)^2 + 3(-3|((-3x - 3)/(2x + 2)) + 3|) - 4) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3sin(2\u03c0(-3((-1(-3x^2 - 2x) - 3)/(1(-3x^2 - 2x))) - 4) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(-5((1(-1cos(2\u03c0x - 1) + 3) + 1)/(-3(-1cos(2\u03c0x - 1) + 3) - 2)) - 1)^2 - 5(-5((1(-1cos(2\u03c0x - 1) + 3) + 1)/(-3(-1cos(2\u03c0x - 1) + 3) - 2)) - 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3|(2sin(1\u03c0((3x - 3)/(2x + 3)) + 3) - 2)| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3(-2cos(3\u03c0(3|x - 2| - 3) - 1) - 1)^2 + 3(-2cos(3\u03c0(3|x - 2| - 3) - 1) - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "178", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(-2(-2cos(3\u03c0x - 2) - 2) + 3) + 1)/(3(-2(-2cos(3\u03c0x - 2) - 2) + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|(1sin(2\u03c0(4(-1x^2 + 5x) - 4) + 1) + 1) + 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "479", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2cos(2\u03c0((-2(-2(-3x^2 + 5x + 3) - 2))/(1(-2(-3x^2 + 5x + 3) - 2) + 3)) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (3(3cos(2\u03c0(2|(-2x^2 + 5)| - 2) + 1) - 1) - 1)/(-2(3cos(2\u03c0(2|(-2x^2 + 5)| - 2) + 1) - 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "218", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 4(3sin(3\u03c0((2x - 2)/(1x - 2))) - 2)^2 - 2(3sin(3\u03c0((2x - 2)/(1x - 2))) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1cos(1\u03c0(-4(2((-1x + 3)/(1x - 2)) + 2)^2 + 4(2((-1x + 3)/(1x - 2)) + 2) - 4) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(3\u03c0(-2(1((1x)/(3x + 3))^2 - 3) + 2) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(-1sin(3\u03c0((1x + 1)/(-1x + 2)) - 1) - 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3((-2(-3|(-3cos(1\u03c0x + 2) + 3) - 1| - 1))/(1(-3|(-3cos(1\u03c0x + 2) + 3) - 1| - 1) + 2)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(2cos(3\u03c0((2(1x - 5) + 2)/(-2(1x - 5) + 3)) - 3) - 2) + 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(1|(-3cos(1\u03c0x) + 3) - 3| + 2)^2 - 5(1|(-3cos(1\u03c0x) + 3) - 3| + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1(-2(2cos(2\u03c0(-3|x - 2| - 3) - 1) + 2)^2 - 3(2cos(2\u03c0(-3|x - 2| - 3) - 1) + 2) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2sin(2\u03c0(-3(1|(2x^2 - 4x + 1) - 1| - 3) - 3) + 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "250", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-2(1(-2(1cos(2\u03c0x) + 3) + 5)^2 + 2(-2(1cos(2\u03c0x) + 3) + 5) + 2) + 2)/(1(1(-2(1cos(2\u03c0x) + 3) + 5)^2 + 2(-2(1cos(2\u03c0x) + 3) + 5) + 2) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2e^(2(-1|(-4x^2 + 1x + 1)|)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1sin(3\u03c0(-4(1|x + 3| - 3) - 1) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5((1(1sin(2\u03c0(3|x - 3| - 3) - 2) + 3) + 3)/(-2(1sin(2\u03c0(3|x - 3| - 3) - 2) + 3) - 2)) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2|((1(2sin(1\u03c0x + 3) - 1) - 2)/(-2(2sin(1\u03c0x + 3) - 1) + 3)) - 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(-1(2cos(2\u03c0x + 1) - 2)^2 + 2(2cos(2\u03c0x + 1) - 2) + 5) + 1)/(-2(-1(2cos(2\u03c0x + 1) - 2)^2 + 2(2cos(2\u03c0x + 1) - 2) + 5) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(-3cos(3\u03c0(-1(2|x + 3| - 3)^2 - 3(2|x + 3| - 3) - 1) - 2) + 3))/(-3(-3cos(3\u03c0(-1(2|x + 3| - 3)^2 - 3(2|x + 3| - 3) - 1) - 2) + 3) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2sin(3\u03c0(-1((1(3x - 3) - 2)/(-2(3x - 3) - 1))^2 - 5((1(3x - 3) - 2)/(-2(3x - 3) - 1)) + 2) - 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(-1(-3cos(1\u03c0((-1x)/(-3x + 4)) - 3) + 3) - 3) + 2| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1|(1(-2cos(3\u03c0((2x - 2)/(3x - 2))) - 3)^2 - 3(-2cos(3\u03c0((2x - 2)/(3x - 2))) - 3) + 4) + 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4(-1sin(3\u03c0(-1|x| + 2) - 1) + 1)^2 - 4(-1sin(3\u03c0(-1|x| + 2) - 1) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-3(-3(1cos(3\u03c0(-2x + 1) + 3) + 3)^2 - 2))/(-1(-3(1cos(3\u03c0(-2x + 1) + 3) + 3)^2 - 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(3|((2(-2cos(3\u03c0x + 1)) - 2)/(1(-2cos(3\u03c0x + 1)) - 3))| + 2) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "113", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2|(1cos(3\u03c0((-3(2x + 3))/(3(2x + 3) + 2)) + 3) - 3) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1((-2(-1|(2sin(2\u03c0x - 3) + 3) + 2| - 2) + 3)/(3(-1|(2sin(2\u03c0x - 3) + 3) + 2| - 2) + 2))^2 - 4((-2(-1|(2sin(2\u03c0x - 3) + 3) + 2| - 2) + 3)/(3(-1|(2sin(2\u03c0x - 3) + 3) + 2| - 2) + 2)) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2((1(1sin(3\u03c0x - 3) - 1) - 3)/(-3(1sin(3\u03c0x - 3) - 1) - 3))^2 - 1((1(1sin(3\u03c0x - 3) - 1) - 3)/(-3(1sin(3\u03c0x - 3) - 1) - 3)) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 4(3sin(2\u03c0(3x - 5) + 2) - 2)^2 - 5(3sin(2\u03c0(3x - 5) + 2) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|(-2sin(3\u03c0(4x^2 + 5x + 2) - 1) - 1)| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "492", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2(4(-1cos(1\u03c0x) - 3) + 4)^2 + 4(4(-1cos(1\u03c0x) - 3) + 4) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1sin(2\u03c0(1(-2x + 5)^2 + 2(-2x + 5) + 5) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "475", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1cos(2\u03c0(4(-3x + 2)^2 - 3(-3x + 2) + 3)) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "229", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(3cos(1\u03c0(-4((1x + 3)/(1x - 2)))) + 1)^2 - 3(3cos(1\u03c0(-4((1x + 3)/(1x - 2)))) + 1) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(-3sin(1\u03c0(1x + 5) + 3) - 2) + 3)/(-2(-3sin(1\u03c0(1x + 5) + 3) - 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(-1(-1e^(2(-3|x - 2|)) + 1)^2 - 2(-1e^(2(-3|x - 2|)) + 1) - 4) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(2cos(3\u03c0(-3|(-3x^2 - 1x + 1)| + 3) + 3) - 2) - 2)/(1(2cos(3\u03c0(-3|(-3x^2 - 1x + 1)| + 3) + 3) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "482", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-2(-2|(2x^2 - 3x + 1) + 3| - 2))/(-3(-2|(2x^2 - 3x + 1) + 3| - 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "307", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-1(-2(1e^(-2(-2|x + 3| + 3)) - 3)^2 + 4(1e^(-2(-2|x + 3| + 3)) - 3) - 5) + 2)/(-3(-2(1e^(-2(-2|x + 3| + 3)) - 3)^2 + 4(1e^(-2(-2|x + 3| + 3)) - 3) - 5)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "157", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(2sin(3\u03c0(-1|x - 3|) + 3) + 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4(-2e^(-1(1|(-1x^2 + 2x - 2) + 3| + 1)) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (2(-1(-2sin(1\u03c0x + 2) - 3) - 3) + 2)/(-2(-1(-2sin(1\u03c0x + 2) - 3) - 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(3\u03c0(1((2(-1x^2 + 5x + 2) - 2)/(1(-1x^2 + 5x + 2) + 3)) + 3) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|((-3(-2(1x^2 + 1x - 1) - 4))/(1(-2(1x^2 + 1x - 1) - 4))) + 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "149", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -5(5((1(2sin(3\u03c0x + 2) + 3) - 1)/(3(2sin(3\u03c0x + 2) + 3) - 1)) - 1)^2 + 5(5((1(2sin(3\u03c0x + 2) + 3) - 1)/(3(2sin(3\u03c0x + 2) + 3) - 1)) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5(-2cos(3\u03c0(-1|(2x - 4) + 3| - 3) - 2) - 3)^2 + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2(1|(1(-2cos(2\u03c0x + 2) + 1)^2 + 3)| + 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(3|(-2x^2 - 3x + 4) + 2| - 2))/(3(3|(-2x^2 - 3x + 4) + 2| - 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-2sin(1\u03c0(5x^2 - 1x - 4) - 1) + 1) + 3)/(-3(-2sin(1\u03c0(5x^2 - 1x - 4) - 1) + 1)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "211", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(-5(1cos(2\u03c0((-3x + 1)/(-1x - 2)) - 1) + 1)^2 + 1(1cos(2\u03c0((-3x + 1)/(-1x - 2)) - 1) + 1)) + 3| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(3sin(1\u03c0((1(4x + 3) - 1)/(-1(4x + 3) - 3))))^2 - 3(3sin(1\u03c0((1(4x + 3) - 1)/(-1(4x + 3) - 3)))) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3(3sin(3\u03c0(1(1|x - 2| - 1)^2 - 5(1|x - 2| - 1)) + 3)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(1(1sin(2\u03c0(1x - 2) + 1) + 2)^2 + 4(1sin(2\u03c0(1x - 2) + 1) + 2) + 4) - 3| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(3|(1sin(2\u03c0(-4x + 2)) - 1) + 1| + 1) + 3)/(-2(3|(1sin(2\u03c0(-4x + 2)) - 1) + 1| + 1) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "320", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (1(3cos(3\u03c0(-4x^2 - 5x + 4) - 2) - 3) + 1)/(-1(3cos(3\u03c0(-4x^2 - 5x + 4) - 2) - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "130", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2cos(1\u03c0(-2((-3(4x^2 + 5x - 1) + 3)/(2(4x^2 + 5x - 1) + 1)) - 5) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(1\u03c0(1(1|x| - 2) - 4) - 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(1|(4(1sin(1\u03c0x) - 2)) - 1| + 1) + 1)/(-2(1|(4(1sin(1\u03c0x) - 2)) - 1| + 1) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3|(4(1cos(1\u03c0x - 3) + 3) + 5) + 2| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(1cos(3\u03c0(-1|((1x)/(-2x - 2)) - 1| + 1)) + 1)^2 + 5(1cos(3\u03c0(-1|((1x)/(-2x - 2)) - 1| + 1)) + 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5(-1sin(3\u03c0(1|x|) + 3) + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|((-3(1sin(1\u03c0x - 2) + 1) - 1)/(-3(1sin(1\u03c0x - 2) + 1) - 2)) + 3| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(3cos(2\u03c0(3|x - 2| + 2) + 2) - 3)^2 - 5(3cos(2\u03c0(3|x - 2| + 2) + 2) - 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2sin(3\u03c0((3(5(2|x - 2| - 3) + 5) - 2)/(2(5(2|x - 2| - 3) + 5))) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1|(-5(3cos(2\u03c0x) - 1) - 5) + 2| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(3(-3sin(2\u03c0x) - 3) - 1) - 2| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(2(-2e^(1(3|x + 3|)) - 3)^2 + 1(-2e^(1(3|x + 3|)) - 3) - 3) + 3)/(2(2(-2e^(1(3|x + 3|)) - 3)^2 + 1(-2e^(1(3|x + 3|)) - 3) - 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(2\u03c0(-3(3|(1x + 2)| - 3)^2 + 3(3|(1x + 2)| - 3) - 1) + 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "477", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(2(1sin(3\u03c0x - 2) - 3) + 2)^2 - 2(2(1sin(3\u03c0x - 2) - 3) + 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3sin(3\u03c0(-4(1x^2 + 3x) - 2) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "252", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2(-3sin(3\u03c0(5x^2 + 1) - 3)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "244", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1cos(3\u03c0(3(-4x + 1)^2 + 3(-4x + 1) - 3) - 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "222", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3cos(3\u03c0(-2|(-3x + 2) + 2| + 3) + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3|(-2sin(2\u03c0(-5((3x - 2)/(1x - 3))^2 - 3((3x - 2)/(1x - 3)) - 5)) - 3) + 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3cos(1\u03c0((-1(5(-5x^2 - 2x + 3)) - 1)/(-1(5(-5x^2 - 2x + 3)) + 2))) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(1\u03c0(1|(-5x - 2) + 3| + 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2((3(-3cos(3\u03c0x + 3) + 2) - 1)/(-3(-3cos(3\u03c0x + 3) + 2) - 3))^2 - 5((3(-3cos(3\u03c0x + 3) + 2) - 1)/(-3(-3cos(3\u03c0x + 3) + 2) - 3)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1|(-2sin(1\u03c0((-2x - 2)/(-2x - 1)) + 1) + 2) - 3| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(-1(-3(3cos(3\u03c0x + 3))^2 + 1(3cos(3\u03c0x + 3)) + 4) + 4) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(1((1(1sin(2\u03c0x - 3) - 1) + 3)/(-3(1sin(2\u03c0x - 3) - 1) + 2))^2 - 4((1(1sin(2\u03c0x - 3) - 1) + 3)/(-3(1sin(2\u03c0x - 3) - 1) + 2)) + 3)| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1cos(2\u03c0(-1((1x - 1)/(-1x + 2)) + 4) - 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3(4(-2sin(2\u03c0x + 1) - 2) - 4)^2 - 1(4(-2sin(2\u03c0x + 1) - 2) - 4) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(3\u03c0((1(-3|x + 2| + 3) - 3)/(-2(-3|x + 2| + 3) + 3)) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(4(2cos(3\u03c0x + 2) - 1)^2 + 5(2cos(3\u03c0x + 2) - 1) - 2) - 1| + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "235", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2cos(1\u03c0((3(-2(-2x + 4)^2 + 5(-2x + 4)) - 1)/(-1(-2(-2x + 4)^2 + 5(-2x + 4)) - 3))) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(-3e^(-2(-1x - 2))) - 3)/(1(-3e^(-2(-1x - 2))) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(-2cos(2\u03c0(4x^2 - 2x + 5) + 2) - 1) + 2)/(2(-2cos(2\u03c0(4x^2 - 2x + 5) + 2) - 1) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "336", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|(-3cos(2\u03c0(4((-2x - 1)/(2x - 1)) - 1) + 2) - 1) - 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3|(-1sin(2\u03c0((-3(2x - 1))/(-1(2x - 1) - 1)) + 2)) - 2| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (1(-2cos(3\u03c0(1(5x - 5)^2 - 2(5x - 5) - 5) - 3) + 2) + 1)/(1(-2cos(3\u03c0(1(5x - 5)^2 - 2(5x - 5) - 5) - 3) + 2) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "221", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3|(2sin(1\u03c0((3x - 3)/(-2x + 3)) + 3)) + 3| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(1(1|(-3sin(3\u03c0x) + 3) - 1| - 1)^2 + 3(1|(-3sin(3\u03c0x) + 3) - 1| - 1) - 1) - 1)/(3(1(1|(-3sin(3\u03c0x) + 3) - 1| - 1)^2 + 3(1|(-3sin(3\u03c0x) + 3) - 1| - 1) - 1)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1|((3(-3(1e^(2x))^2 - 4(1e^(2x)) + 2) + 1)/(-2(-3(1e^(2x))^2 - 4(1e^(2x)) + 2) - 3)) - 2| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 4(-3|(-1(2sin(3\u03c0x + 2))^2 - 2) + 2| + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3(-2cos(1\u03c0(-1|x - 1| - 3) - 1))^2 - 5(-2cos(1\u03c0(-1|x - 1| - 3) - 1)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2(-1e^(1(-3|x - 3|)) - 3)^2 + 2(-1e^(1(-3|x - 3|)) - 3) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(-3sin(2\u03c0((-3x - 1)/(-2x + 1)) - 2) - 1) + 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(3\u03c0((-1(4x + 5) - 3)/(1(4x + 5) - 2))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(3cos(1\u03c0(-2|x + 2| - 3) + 1) - 1)^2 + 2(3cos(1\u03c0(-2|x + 2| - 3) + 1) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (2(-1|(-3(-1sin(3\u03c0x - 3) + 2)^2 - 1(-1sin(3\u03c0x - 3) + 2) + 2) + 2| + 2))/(3(-1|(-3(-1sin(3\u03c0x - 3) + 2)^2 - 1(-1sin(3\u03c0x - 3) + 2) + 2) + 2| + 2) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(3cos(2\u03c0((2x)/(2x + 1))) - 3)| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1(1|((-1x - 1)/(-3x - 3)) - 1| - 1)^2 + 1(1|((-1x - 1)/(-3x - 3)) - 1| - 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(-2sin(2\u03c0(5(2x + 2)^2 - 1) - 3) - 3) - 1)/(-3(-2sin(2\u03c0(5(2x + 2)^2 - 1) - 3) - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "340", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(1\u03c0((3(-5x^2 - 2x - 4) + 3)/(2(-5x^2 - 2x - 4) + 2)) + 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3|(-2(-3cos(3\u03c0x + 3) - 1) - 3) - 1| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(3\u03c0((3(3x^2 - 3x + 1) + 3)/(-2(3x^2 - 3x + 1))) + 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 5((-2(-3cos(1\u03c0(2|x + 3| + 1) - 2) - 1) - 1)/(-1(-3cos(1\u03c0(2|x + 3| + 1) - 2) - 1) - 3))^2 - 4((-2(-3cos(1\u03c0(2|x + 3| + 1) - 2) - 1) - 1)/(-1(-3cos(1\u03c0(2|x + 3| + 1) - 2) - 1) - 3)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3(2sin(2\u03c0(-2|x - 1| - 2) - 3) - 3)^2 + 3(2sin(2\u03c0(-2|x - 1| - 2) - 3) - 3) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(2cos(1\u03c0(2((1x - 3)/(-3x + 4)) - 2) + 1) + 1) - 1| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3cos(1\u03c0(-2|(-1((3x - 1)/(-1x - 3))^2 - 4((3x - 1)/(-1x - 3)) + 3)| - 2) + 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4((2(1sin(1\u03c0x + 2) - 1) - 3)/(-1(1sin(1\u03c0x + 2) - 1) + 2)) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3|((-2(1cos(2\u03c0(-3x - 3) + 3) + 3) + 3)/(3(1cos(2\u03c0(-3x - 3) + 3) + 3) + 3)) - 2| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (1(3|(-2(-3sin(2\u03c0x) + 3)^2 - 2(-3sin(2\u03c0x) + 3) + 2) + 1| + 3))/(-1(3|(-2(-3sin(2\u03c0x) + 3)^2 - 2(-3sin(2\u03c0x) + 3) + 2) + 1| + 3) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1sin(2\u03c0(-3|(-4x + 3)| + 1) + 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3((-2(1|(-1cos(2\u03c0x - 2) + 1)| - 3))/(2(1|(-1cos(2\u03c0x - 2) + 1)| - 3) - 1))^2 - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2((-3(4(1sin(1\u03c0x + 3) + 1) + 4) - 2)/(-1(4(1sin(1\u03c0x + 3) + 1) + 4) - 1))^2 - 3((-3(4(1sin(1\u03c0x + 3) + 1) + 4) - 2)/(-1(4(1sin(1\u03c0x + 3) + 1) + 4) - 1)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(2sin(2\u03c0(-2|(-1x + 5) + 3|) + 1) - 3) - 1)/(1(2sin(2\u03c0(-2|(-1x + 5) + 3|) + 1) - 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(2sin(3\u03c0(2|(-1x^2 + 4x - 2) + 2| - 1) + 3) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "499", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(1cos(3\u03c0((1(3|x + 3| + 1) - 2)/(2(3|x + 3| + 1))) + 1))^2 - 4(1cos(3\u03c0((1(3|x + 3| + 1) - 2)/(2(3|x + 3| + 1))) + 1)) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(-1(1(1sin(3\u03c0x - 3) - 3))^2 + 3(1(1sin(3\u03c0x - 3) - 3)) + 2) - 1)/(1(-1(1(1sin(3\u03c0x - 3) - 3))^2 + 3(1(1sin(3\u03c0x - 3) - 3)) + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -5(1|(-1(3cos(3\u03c0x + 1) + 1)^2 - 2) - 3| - 1) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3sin(2\u03c0(-1|((-1x - 3)/(3x + 3)) + 1| + 3)) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2|(-1cos(1\u03c0(-1x - 2) + 2) + 3) + 2| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3cos(1\u03c0(3((2x + 3)/(-3x + 2)) - 5) + 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1(2sin(1\u03c0((-1x - 2)/(-1x + 2)) - 2)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(2cos(2\u03c0(-3((2x + 3)/(2x + 2)) + 5) - 3) - 1)^2 + 5(2cos(2\u03c0(-3((2x + 3)/(2x + 2)) + 5) - 3) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (2(3|(-5(-3cos(2\u03c0x - 1) - 2) + 5) - 2| + 1) - 1)/(3(3|(-5(-3cos(2\u03c0x - 1) - 2) + 5) - 2| + 1) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1cos(2\u03c0(4(-1x + 4)^2 + 3(-1x + 4) + 3)) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1cos(1\u03c0(-3(-2((3x)/(2x + 1)) + 2)^2 - 4(-2((3x)/(2x + 1)) + 2) + 4) - 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-2(-2sin(1\u03c0(-1x^2 - 4x - 2) + 3) + 2) + 3)/(-2(-2sin(1\u03c0(-1x^2 - 4x - 2) + 3) + 2) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "111", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(1|(2(-2x^2 - 4x + 3) + 3) + 1|)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "285", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(2cos(2\u03c0(1x^2 + 3) - 3)) - 3)/(-3(2cos(2\u03c0(1x^2 + 3) - 3)) + 4) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "184", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3cos(2\u03c0(-3|((-3x + 3)/(-3x + 4)) + 1| - 2) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(1\u03c0(4((-2x - 3)/(-3x - 1))^2 - 5((-2x - 3)/(-3x - 1)) - 1)) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2sin(1\u03c0(-1((-3x - 2)/(2x + 2))^2 + 5((-3x - 2)/(2x + 2)) + 5) - 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(2sin(3\u03c0((3x - 1)/(3x + 2))) + 1) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (3(-1|(-3cos(3\u03c0(-2x - 1)) + 3) - 1| + 1) + 3)/(-1(-1|(-3cos(3\u03c0(-2x - 1)) + 3) - 1| + 1) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(-1(-3cos(1\u03c0x - 2) + 2)^2 - 1(-3cos(1\u03c0x - 2) + 2) + 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2sin(2\u03c0((2(-1x - 3))/(1(-1x - 3) + 1)) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3|((1(-2(3cos(1\u03c0x + 2) - 1) + 3) - 1)/(2(-2(3cos(1\u03c0x + 2) - 1) + 3) - 1)) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0(-3|(3x - 4) + 1| - 2) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(3\u03c0(-1|(-3((2x + 1)/(-2x + 3)) + 4) - 1| - 1) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(1\u03c0(-2(-1|x + 2| - 2)^2 + 4(-1|x + 2| - 2) - 2) - 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "316", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|(-4(-2sin(3\u03c0(-1x^2 - 1x)) + 3)) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "279", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(-2cos(2\u03c0((2x - 3)/(-3x - 3)) - 3) - 3)^2 - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2cos(3\u03c0(-2|(-4x + 4) - 3|) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "479", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1(-2sin(2\u03c0((-2x - 2)/(1x - 3))) - 1)^2 - 2(-2sin(2\u03c0((-2x - 2)/(1x - 3))) - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (2(2(1(3cos(2\u03c0x - 3))^2 - 2(3cos(2\u03c0x - 3)) + 1) + 2))/(2(2(1(3cos(2\u03c0x - 3))^2 - 2(3cos(2\u03c0x - 3)) + 1) + 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(1(-1cos(2\u03c0x - 1) + 1)) - 2)/(-2(1(-1cos(2\u03c0x - 1) + 1)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-2(3cos(3\u03c0(-3|x - 2| + 3) - 1) + 2) - 2)/(1(3cos(3\u03c0(-3|x - 2| + 3) - 1) + 2) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1sin(2\u03c0((1(-1|(-1x + 5) - 2| - 2))/(3(-1|(-1x + 5) - 2| - 2))) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5((2(3(-2sin(3\u03c0x) + 2)^2 - 3(-2sin(3\u03c0x) + 2)) - 1)/(-1(3(-2sin(3\u03c0x) + 2)^2 - 3(-2sin(3\u03c0x) + 2)) - 3)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1(-1cos(2\u03c0(-1|x + 1| - 3)) + 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3|(2sin(3\u03c0((3x + 1)/(-1x - 1)) + 3) + 3) + 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -5((-1(-2|(-1cos(1\u03c0x) - 2) + 2| - 1) - 2)/(-2(-2|(-1cos(1\u03c0x) - 2) + 2| - 1) + 3))^2 - 3((-1(-2|(-1cos(1\u03c0x) - 2) + 2| - 1) - 2)/(-2(-2|(-1cos(1\u03c0x) - 2) + 2| - 1) + 3)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -4(3cos(1\u03c0(2|x + 3| + 1) + 3) - 3)^2 + 2(3cos(1\u03c0(2|x + 3| + 1) + 3) - 3) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2((3(-1cos(1\u03c0x + 2) - 1))/(2(-1cos(1\u03c0x + 2) - 1) - 1)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1((-3(-1sin(1\u03c0x - 1)))/(-2(-1sin(1\u03c0x - 1)) + 3)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3cos(3\u03c0(2|(2x - 4) + 1| + 2) - 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2(-2cos(1\u03c0((1x - 1)/(-1x + 3)) + 1) + 2)^2 + 5(-2cos(1\u03c0((1x - 1)/(-1x + 3)) + 1) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(1(3sin(3\u03c0(-1|x + 1| - 1) + 1) + 2)^2 + 3) - 1)/(-3(1(3sin(3\u03c0(-1|x + 1| - 1) + 1) + 2)^2 + 3) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1sin(2\u03c0((-2(3x + 2) - 3)/(-2(3x + 2) - 1)) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(1(2sin(1\u03c0x - 2) - 1) - 3)^2 + 3(1(2sin(1\u03c0x - 2) - 1) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(-3|(-3cos(3\u03c0(2x^2 + 4x - 4) - 2) + 3) - 1| - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "471", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(1sin(2\u03c0(-1x^2 + 2x - 2) - 3) + 2) + 2| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "203", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3cos(3\u03c0(2|((2(5x - 4))/(1(5x - 4) + 1)) + 3| + 3) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(-1(-2cos(3\u03c0x + 1) - 2) + 1) - 3)/(-3(-1(-2cos(3\u03c0x + 1) - 2) + 1) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2((1(-5(-1sin(1\u03c0x + 2) + 2)^2 - 3(-1sin(1\u03c0x + 2) + 2) - 5) - 3)/(3(-5(-1sin(1\u03c0x + 2) + 2)^2 - 3(-1sin(1\u03c0x + 2) + 2) - 5) - 1)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1(2e^(-2(2|x + 3| - 3)) - 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2|(2sin(2\u03c0(3x^2 - 5x + 1) + 2) - 3) + 1| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "227", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3|(5(-1(1sin(1\u03c0x + 1)) + 1)^2 - 4(-1(1sin(1\u03c0x + 1)) + 1) - 5) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2(5(2sin(1\u03c0x - 2) + 1)^2 + 5(2sin(1\u03c0x - 2) + 1) + 1) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3|(4(1sin(2\u03c0(2x - 1) - 2) + 2)^2 - 1(1sin(2\u03c0(2x - 1) - 2) + 2) + 3) + 2| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2sin(2\u03c0(-1(-1x^2 - 2x + 5)) + 2) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3(-3|(1sin(3\u03c0x + 2) + 3)| + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2sin(3\u03c0(1(3|x + 1| + 2)^2 + 4(3|x + 1| + 2) - 4) + 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "476", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -4(3cos(2\u03c0(-1x^2 - 4x + 4) + 2) + 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "211", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1(1|(-2sin(2\u03c0(1x^2 - 5x + 2)) + 3) - 2| + 3) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "496", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2(2sin(2\u03c0(1((-1x - 2)/(-2x))^2 + 1((-1x - 2)/(-2x)) - 1) - 3) + 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3sin(1\u03c0(-2|(4x^2 + 1)| + 2)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "257", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4(-1cos(1\u03c0((3(3x^2 - 5x - 4) - 3)/(1(3x^2 - 5x - 4))) + 1) - 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3(3|((-1(1cos(2\u03c0x) - 1) + 1)/(-3(1cos(2\u03c0x) - 1) + 4)) + 1| + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(-2sin(1\u03c0(5x^2 - 4x) + 2) + 3) - 3)/(1(-2sin(1\u03c0(5x^2 - 4x) + 2) + 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "330", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1cos(1\u03c0(3|((2(2x^2 - 3x - 1))/(2(2x^2 - 3x - 1) - 1)) - 3| + 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(-2|(-2sin(3\u03c0x - 1) + 3) - 2| + 1) + 1)/(-1(-2|(-2sin(3\u03c0x - 1) + 3) - 2| + 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(3|(2cos(2\u03c0x + 3) - 3)| + 2) - 1)/(-3(3|(2cos(2\u03c0x + 3) - 3)| + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3cos(1\u03c0(1(2|x + 2| + 2)^2 + 3(2|x + 2| + 2) + 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "296", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|((2(-3cos(2\u03c0(-1x) + 1)) + 3)/(1(-3cos(2\u03c0(-1x) + 1)))) + 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2sin(3\u03c0(-4((-1x - 2)/(-1x - 1)) - 4) - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1(-4(-2cos(3\u03c0x - 1) + 2) - 3)^2 - 5(-4(-2cos(3\u03c0x - 1) + 2) - 3) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(5(1(-2sin(1\u03c0x) - 3) + 1)^2 + 3(1(-2sin(1\u03c0x) - 3) + 1) - 3) - 3)/(-3(5(1(-2sin(1\u03c0x) - 3) + 1)^2 + 3(1(-2sin(1\u03c0x) - 3) + 1) - 3) + 4) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(4(-2sin(3\u03c0(-5x^2 - 3x) - 1) + 3) + 1) + 3| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "239", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 5(-2cos(1\u03c0(-5x + 1) + 2))^2 - 2(-2cos(1\u03c0(-5x + 1) + 2)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(2|(1cos(1\u03c0x - 3)) + 2|)^2 - 3(2|(1cos(1\u03c0x - 3)) + 2|) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(1\u03c0(5(-1|x - 2| + 3)^2 + 3(-1|x - 2| + 3) + 5) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "230", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0(-5(1|((-1x - 3)/(3x - 2)) - 1|) - 5) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(-2(1cos(1\u03c0(3x + 5) + 3))^2 - 2(1cos(1\u03c0(3x + 5) + 3)) + 2))/(-1(-2(1cos(1\u03c0(3x + 5) + 3))^2 - 2(1cos(1\u03c0(3x + 5) + 3)) + 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(2\u03c0(-3(-1|((-2x - 2)/(-3x)) - 2|) - 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1sin(1\u03c0(2(3|x + 1| - 1)^2 - 4(3|x + 1| - 1) - 3)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "490", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (2(1|(-1(-1sin(3\u03c0x + 3))^2 - 1(-1sin(3\u03c0x + 3)) - 4) + 1| + 1) + 3)/(-3(1|(-1(-1sin(3\u03c0x + 3))^2 - 1(-1sin(3\u03c0x + 3)) - 4) + 1| + 1) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2cos(2\u03c0(-3|((1x + 3)/(2x + 2)) + 1| + 1) - 1) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3sin(2\u03c0((2(3|x + 2| - 2) - 3)/(-1(3|x + 2| - 2) - 1)) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2(3sin(1\u03c0(2|((-3x + 2)/(3x + 1)) + 2| - 2) - 3) + 1) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1(-4((3(2sin(2\u03c0x) - 1) + 2)/(2(2sin(2\u03c0x) - 1) - 1))^2 - 1((3(2sin(2\u03c0x) - 1) + 2)/(2(2sin(2\u03c0x) - 1) - 1)) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1sin(1\u03c0(-1|((-3x - 2)/(-2x + 3)) - 1|) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(-3|(-2cos(2\u03c0x + 3) + 2)| + 2)^2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3|(1cos(1\u03c0(1((1x - 1)/(2x + 3)) + 4) + 1) + 2) - 1| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(3|(-3cos(2\u03c0(4x + 5) + 3)) + 3| - 3) + 2)/(-1(3|(-3cos(2\u03c0(4x + 5) + 3)) + 3| - 3)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2cos(2\u03c0(-4(2|(1x^2 - 5) + 1|) + 3) + 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 4(1(1sin(1\u03c0((3x + 1)/(-2x + 1)) - 2) - 2) + 3)^2 + 3(1(1sin(1\u03c0((3x + 1)/(-2x + 1)) - 2) - 2) + 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -4(-1cos(3\u03c0(-2|x - 3| + 2) - 2)) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2|(-5(-2cos(1\u03c0x + 3) + 1) + 2) + 3| + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(2\u03c0(-2((-1x - 3)/(2x - 1)) - 4)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(1cos(1\u03c0(3|((-2x)/(-2x + 3)) - 1|)) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -5(1sin(2\u03c0(-1x + 5) + 3) + 3)^2 + 2(1sin(2\u03c0(-1x + 5) + 3) + 3) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(3\u03c0(-3|(2((1x + 2)/(-1x))^2 - 5((1x + 2)/(-1x)) + 2) - 1|) - 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (3(-5(3cos(2\u03c0x + 3)) + 3) + 2)/(1(-5(3cos(2\u03c0x + 3)) + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|(2(2sin(3\u03c0x + 1)) - 2) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4((3(-5(-3|x - 1|)^2 - 1(-3|x - 1|) + 2))/(1(-5(-3|x - 1|)^2 - 1(-3|x - 1|) + 2))) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-1(-1|(2cos(3\u03c0x - 1) + 1) + 3| - 3))/(-2(-1|(2cos(3\u03c0x - 1) + 1) + 3| - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (3(3sin(1\u03c0(-3|x + 3| - 2) + 2)) + 3)/(1(3sin(1\u03c0(-3|x + 3| - 2) + 2)) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-2sin(1\u03c0((3(1x^2 - 3x + 1) + 3)/(1(1x^2 - 3x + 1)))) - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2sin(2\u03c0((2(1|x| - 1) + 1)/(-1(1|x| - 1) - 1)) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3sin(2\u03c0(1(-2|(5x - 2) + 1| + 2)^2 + 2(-2|(5x - 2) + 1| + 2) + 1) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "197", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1(-3((3(3sin(1\u03c0x) + 1) + 2)/(3(3sin(1\u03c0x) + 1) - 2)) + 3)^2 - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2cos(3\u03c0(-2((1x + 2)/(-3x + 2)) + 1) - 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(-1|(1cos(3\u03c0(-1x^2 + 3x - 3) + 1) + 2) - 1| + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "555", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1|(-4(3sin(3\u03c0x + 2) - 2)^2 - 1(3sin(3\u03c0x + 2) - 2) + 5) + 1| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(-1(-1sin(3\u03c0(-1x^2 + 4) + 1) - 3) + 4)| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "565", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(-1cos(1\u03c0(-1((3x - 3)/(-3x - 1))^2 + 4((3x - 3)/(-3x - 1)) + 1) + 1) + 3)| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2(-3cos(2\u03c0(-2|x| + 2) - 1) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1(2sin(1\u03c0(2|x + 3| + 3) - 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3|(2cos(1\u03c0(3x - 5) - 2)) + 3| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -5(-3sin(3\u03c0(-3(-1|x + 2| - 2)^2 - 5(-1|x + 2| - 2) + 5) - 3) - 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "268", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1cos(3\u03c0(-4(-3|x - 3| - 3) + 2)) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(2cos(1\u03c0(-2|((-1x - 3)/(-1x - 2)) + 3| + 2) - 2) - 1) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2cos(1\u03c0(-3(5x^2 + 5x - 2) - 1)) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "502", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3cos(2\u03c0((-3(1x^2 + 4x + 1) + 1)/(-2(1x^2 + 4x + 1) - 1)) - 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3cos(3\u03c0(-4(1(-1|x - 2| - 3)^2 - 2(-1|x - 2| - 3) - 1) + 3) - 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "254", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(1|(-1sin(3\u03c0(-1x^2 - 2x) + 2) - 3)| + 2) + 2)/(-1(1|(-1sin(3\u03c0(-1x^2 - 2x) + 2) - 3)| + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "171", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (3(5(3sin(3\u03c0(1x - 1) + 2) + 2)^2 - 1(3sin(3\u03c0(1x - 1) + 2) + 2)) + 3)/(3(5(3sin(3\u03c0(1x - 1) + 2) + 2)^2 - 1(3sin(3\u03c0(1x - 1) + 2) + 2)) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(3(3cos(3\u03c0x - 2) - 3)^2 - 2(3cos(3\u03c0x - 2) - 3) - 3) - 2)/(1(3(3cos(3\u03c0x - 2) - 3)^2 - 2(3cos(3\u03c0x - 2) - 3) - 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(5(2(1cos(1\u03c0x - 3) + 1)^2 + 3(1cos(1\u03c0x - 3) + 1) - 4) - 4) + 2)/(1(5(2(1cos(1\u03c0x - 3) + 1)^2 + 3(1cos(1\u03c0x - 3) + 1) - 4) - 4) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(2\u03c0(1((3x - 1)/(-3x + 4))^2 - 3((3x - 1)/(-3x + 4)) - 1) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-1(-1cos(3\u03c0(-2|x + 1| - 1) - 1) + 2))/(-3(-1cos(3\u03c0(-2|x + 1| - 1) - 1) + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(-3(1sin(2\u03c0((2x + 1)/(1x + 1)) - 3) + 1) + 5)| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2(3(-1|(2cos(1\u03c0x - 3) + 2) + 1| + 2) + 3)^2 + 1(3(-1|(2cos(1\u03c0x - 3) + 2) + 1| + 2) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(2cos(2\u03c0(-1x^2 - 1x - 5))) + 1)/(1(2cos(2\u03c0(-1x^2 - 1x - 5)))) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 5((-3(1|x| + 2))/(-1(1|x| + 2)))^2 + 4((-3(1|x| + 2))/(-1(1|x| + 2))) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(-3cos(3\u03c0((-2(-2|x - 3| + 2) - 2)/(-3(-2|x - 3| + 2) + 2)) + 2) + 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3|(-2cos(3\u03c0(1(5x)^2 + 5(5x) + 2) + 2) + 2) - 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3cos(2\u03c0(3(2|(-3x) - 2|)^2 + 2(2|(-3x) - 2|) - 3) + 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "182", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|((1(-1(3sin(2\u03c0x + 1) - 1)^2 + 1(3sin(2\u03c0x + 1) - 1) + 4) - 2)/(-2(-1(3sin(2\u03c0x + 1) - 1)^2 + 1(3sin(2\u03c0x + 1) - 1) + 4) + 3)) - 1| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1cos(2\u03c0(4((2(2|x - 3| - 3))/(2(2|x - 3| - 3) + 2)) + 3)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4(2|(1cos(1\u03c0((-3x + 1)/(-2x - 1)) - 3) - 2) + 1| - 1) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-1(-1(3(-1cos(3\u03c0x + 1) + 1)^2 - 2(-1cos(3\u03c0x + 1) + 1) - 5)) + 3)/(-1(-1(3(-1cos(3\u03c0x + 1) + 1)^2 - 2(-1cos(3\u03c0x + 1) + 1) - 5)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (1(2|(-3(-2sin(1\u03c0x + 3) - 3)^2)| - 3) - 2)/(1(2|(-3(-2sin(1\u03c0x + 3) - 3)^2)| - 3) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(-3cos(3\u03c0((2(-1x - 5) - 3)/(-3(-1x - 5) - 3)) - 3) + 3) - 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(-3sin(1\u03c0(-5x - 1) - 2) + 3)| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2|(-1(1cos(2\u03c0(-2x^2 - 3x + 4))) + 1) + 3| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "281", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 4((2(-2sin(2\u03c0x + 3)) + 1)/(-1(-2sin(2\u03c0x + 3)) + 2)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (1(-1sin(3\u03c0(1|x - 2| + 3) - 2) - 3) + 2)/(3(-1sin(3\u03c0(1|x - 2| + 3) - 2) - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|(2cos(1\u03c0(4x^2 + 2x) - 2) + 3) - 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "285", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(2|(1(2sin(1\u03c0x + 1) + 1)^2 + 2(2sin(1\u03c0x + 1) + 1) - 3) - 3| + 3) + 3)/(1(2|(1(2sin(1\u03c0x + 1) + 1)^2 + 2(2sin(1\u03c0x + 1) + 1) - 3) - 3| + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3cos(1\u03c0((3(4(1|x - 1|) + 5))/(2(4(1|x - 1|) + 5)))) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-3(1e^(-1(-4x + 5)) - 2) + 1)/(2(1e^(-1(-4x + 5)) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1(1cos(2\u03c0((-3x + 1)/(-1x - 1))) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|(-1(3sin(2\u03c0x)) - 3) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(-5(1x^2 + 5x - 1) + 3))/(3(-5(1x^2 + 5x - 1) + 3)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "246", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3|(-1cos(3\u03c0(-5(-1x^2 - 1x + 2) + 3) - 1)) - 3| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "255", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1sin(2\u03c0(-3|(-5x^2 + 3) - 1|) - 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "491", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3(-1|(-1sin(3\u03c0x + 1) + 3)| + 1) + 3)/(-3(-1|(-1sin(3\u03c0x + 1) + 3)| + 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3(1sin(2\u03c0(5x^2 - 5) + 2) - 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(-1cos(1\u03c0(1((-2x - 1)/(2x - 1))^2 + 1((-2x - 1)/(2x - 1))) + 1)) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(1|(-2(-3sin(1\u03c0x - 1) - 3)) - 2| - 2)^2 - 3(1|(-2(-3sin(1\u03c0x - 1) - 3)) - 2| - 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4((3(-2e^(-1(3x - 5)) - 1))/(-1(-2e^(-1(3x - 5)) - 1) - 2))^2 - 2((3(-2e^(-1(3x - 5)) - 1))/(-1(-2e^(-1(3x - 5)) - 1) - 2)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3cos(2\u03c0(-5((-2x + 2)/(2x - 1)) + 4) + 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-3sin(3\u03c0(5x - 5) + 1) - 3)^2 - 2(-3sin(3\u03c0(5x - 5) + 1) - 3) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -5(-2|((2(-2cos(1\u03c0x + 2) + 2) - 2)/(-1(-2cos(1\u03c0x + 2) + 2) - 1))| + 3)^2 + 5(-2|((2(-2cos(1\u03c0x + 2) + 2) - 2)/(-1(-2cos(1\u03c0x + 2) + 2) - 1))| + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 4(2|(2cos(1\u03c0(2x - 3)) - 3) - 2| + 3)^2 - 5(2|(2cos(1\u03c0(2x - 3)) - 3) - 2| + 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(-4((-3(2cos(3\u03c0x - 1) + 3) + 3)/(3(2cos(3\u03c0x - 1) + 3) + 1))^2 - 1((-3(2cos(3\u03c0x - 1) + 3) + 3)/(3(2cos(3\u03c0x - 1) + 3) + 1)) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(3\u03c0(-1(-2|x - 1| - 1) - 5)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(-1(-2sin(1\u03c0x - 3) - 2) + 3) - 2)/(-2(-1(-2sin(1\u03c0x - 3) - 2) + 3) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2(1|(2cos(1\u03c0((3x + 1)/(1x)) - 1) + 3) + 2| + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2|(-1(2sin(1\u03c0((3x + 3)/(-1x)) - 3) + 2)^2 - 2) + 2| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(2\u03c0(-3(-3|x| + 2)^2 + 4(-3|x| + 2) + 1)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "475", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(2sin(1\u03c0((-3x + 2)/(3x)) + 2) - 3) - 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3(-2cos(3\u03c0(3((-2x + 2)/(3x + 1))^2 - 1((-2x + 2)/(3x + 1)) + 5) - 1)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2(5((3(-1sin(1\u03c0x + 2) - 2) + 3)/(-2(-1sin(1\u03c0x + 2) - 2) + 3)) - 4)^2 + 3(5((3(-1sin(1\u03c0x + 2) - 2) + 3)/(-2(-1sin(1\u03c0x + 2) - 2) + 3)) - 4) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(-3cos(2\u03c0((1x + 1)/(-2x + 3)) - 3) + 3)^2 - 4(-3cos(2\u03c0((1x + 1)/(-2x + 3)) - 3) + 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -5(-5(1cos(1\u03c0x + 2) - 1)^2 + 4(1cos(1\u03c0x + 2) - 1) + 5) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(-3(-2cos(2\u03c0x - 1) + 2) + 1) + 2| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1sin(2\u03c0(-5((1x - 1)/(2x - 3))^2 + 4((1x - 1)/(2x - 3)) - 5) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1|((-1(2(1sin(3\u03c0x - 2) - 2)^2 - 5(1sin(3\u03c0x - 2) - 2)) + 3)/(2(2(1sin(3\u03c0x - 2) - 2)^2 - 5(1sin(3\u03c0x - 2) - 2)) + 3))| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3|((2(3cos(1\u03c0x - 3) + 1) - 2)/(-2(3cos(1\u03c0x - 3) + 1) - 1)) - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(3\u03c0(2|((1x + 1)/(-2x + 3))| + 3) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3sin(2\u03c0(-2(2|x - 2| + 2)^2) - 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "252", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1cos(1\u03c0(-1(1x^2 + 3x + 1)) - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "103", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-3(-2(-3|(2x - 5) - 3| + 2)^2 - 3(-3|(2x - 5) - 3| + 2) + 2))/(-1(-2(-3|(2x - 5) - 3| + 2)^2 - 3(-3|(2x - 5) - 3| + 2) + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "136", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (1(-2(3cos(3\u03c0(5x + 3) + 1) + 3)^2 + 4(3cos(3\u03c0(5x + 3) + 1) + 3) - 1) + 2)/(-2(-2(3cos(3\u03c0(5x + 3) + 1) + 3)^2 + 4(3cos(3\u03c0(5x + 3) + 1) + 3) - 1) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3((-1(3sin(3\u03c0x - 1)) - 1)/(-3(3sin(3\u03c0x - 1)) + 4)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1(2|(3cos(3\u03c0x - 3))| - 2)^2 - 4(2|(3cos(3\u03c0x - 3))| - 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 5(5(-1cos(2\u03c0x - 2) - 2)^2 - 1(-1cos(2\u03c0x - 2) - 2) + 5) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(2(3sin(3\u03c0x - 1) + 1) + 4))/(2(2(3sin(3\u03c0x - 1) + 1) + 4) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3cos(3\u03c0(-3|(5(2x^2 + 5) - 4) + 3| - 3) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "425", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-4(-2e^(2(2|x + 1| - 1)))^2 - 4(-2e^(2(2|x + 1| - 1))) + 4) + 3)/(3(-4(-2e^(2(2|x + 1| - 1)))^2 - 4(-2e^(2(2|x + 1| - 1))) + 4) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(3\u03c0((1(3|(-2x^2 - 4x - 1) + 3| - 3) - 3)/(3(3|(-2x^2 - 4x - 1) + 3| - 3) - 1))) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(3\u03c0(-2|((3(5x) - 3)/(-1(5x) + 2)) + 2| + 1) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2sin(3\u03c0(-3(-1|x + 1|) + 2) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1(-2sin(2\u03c0((1x - 2)/(1x)) - 3) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1(-2cos(3\u03c0(4x^2 - 2x + 3) - 3) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "243", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4(1(1|(1sin(2\u03c0x - 2) + 3) - 3| - 2)^2 - 5(1|(1sin(2\u03c0x - 2) + 3) - 3| - 2) + 5) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3sin(3\u03c0(3|((1x - 2)/(-2x - 2)) - 2|) + 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-1(-2sin(1\u03c0(-2|(1x^2 + 2x - 5) + 2| + 3) + 2) - 2) + 3)/(-2(-2sin(1\u03c0(-2|(1x^2 + 2x - 5) + 2| + 3) + 2) - 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "209", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|((-1(1cos(1\u03c0x + 2) - 3) - 2)/(2(1cos(1\u03c0x + 2) - 3) - 1)) + 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(1cos(3\u03c0(-4(-1|x - 2| + 3)^2 - 4(-1|x - 2| + 3) + 2) + 2) + 2) - 1)/(3(1cos(3\u03c0(-4(-1|x - 2| + 3)^2 - 4(-1|x - 2| + 3) + 2) + 2) + 2) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "514", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3(-1sin(2\u03c0(1x^2 + 1x - 1) - 1) + 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "202", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(-3sin(2\u03c0(-2(-1x^2 - 5x + 3) - 4) + 2) + 2) + 2)/(-1(-3sin(2\u03c0(-2(-1x^2 - 5x + 3) - 4) + 2) + 2) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "393", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(-3sin(3\u03c0(-1x + 5) - 2) - 1)^2 - 3(-3sin(3\u03c0(-1x + 5) - 2) - 1) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 4(-2|((-3(-1e^(1x) - 1) - 3)/(1(-1e^(1x) - 1) + 1)) + 1| - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "173", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2cos(2\u03c0(-5(3|x - 3| + 3)^2 + 2(3|x - 3| + 3) - 5) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "218", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-4(2cos(2\u03c0x - 3) - 2) + 5) - 3)/(-2(-4(2cos(2\u03c0x - 3) - 2) + 5) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-1(2(1cos(2\u03c0x - 1) + 1)^2 - 5(1cos(2\u03c0x - 1) + 1) - 4) + 3)/(-2(2(1cos(2\u03c0x - 1) + 1)^2 - 5(1cos(2\u03c0x - 1) + 1) - 4) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -4((-2(-3(1cos(2\u03c0x - 2) + 2) - 1) + 3)/(3(-3(1cos(2\u03c0x - 2) + 2) - 1) - 2))^2 + 2((-2(-3(1cos(2\u03c0x - 2) + 2) - 1) + 3)/(3(-3(1cos(2\u03c0x - 2) + 2) - 1) - 2)) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1cos(2\u03c0(1((1(-1|x - 1| + 3) - 3)/(-1(-1|x - 1| + 3)))^2 + 3((1(-1|x - 1| + 3) - 3)/(-1(-1|x - 1| + 3))) + 4) + 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -4(-5(-3sin(1\u03c0((1x + 1)/(1x + 2)) + 3) + 2) + 4)^2 - 3(-5(-3sin(1\u03c0((1x + 1)/(1x + 2)) + 3) + 2) + 4) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(3sin(1\u03c0((3x + 2)/(1x - 2)) - 1))| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(3\u03c0(-3|(2x + 3) - 3| - 1)) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(-2sin(2\u03c0(-1|(2x^2 - 3x + 4) + 1| + 3) - 2) - 1) - 2)/(3(-2sin(2\u03c0(-1|(2x^2 - 3x + 4) + 1| + 3) - 2) - 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "308", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1|((3(1e^(2(-1x + 5)) + 1) + 1)/(2(1e^(2(-1x + 5)) + 1) + 1))| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(-2cos(2\u03c0((-2x + 1)/(1x)) + 1) + 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(1\u03c0(2(-2|(-5x) - 2| - 3)^2 + 1(-2|(-5x) - 2| - 3) - 2) + 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "195", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(-3(-2e^(2((-3x - 3)/(-1x - 1))) + 3) - 1)^2 + 5(-3(-2e^(2((-3x - 3)/(-1x - 1))) + 3) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2cos(3\u03c0(-3((-1(-4x - 3) + 1)/(2(-4x - 3) + 1))^2 + 2((-1(-4x - 3) + 1)/(2(-4x - 3) + 1)) - 5) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|(3((3(2x^2 + 5x + 3))/(-2(2x^2 + 5x + 3))) - 1) + 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "171", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3(-1(-2sin(1\u03c0((-2x + 2)/(-3x + 2)) + 2) - 2) - 3)^2 + 2(-1(-2sin(1\u03c0((-2x + 2)/(-3x + 2)) + 2) - 2) - 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3sin(3\u03c0((-3(-2x) + 3)/(3(-2x) - 2)) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(2\u03c0(-1|(-3x + 2) + 2| - 2) - 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1|(-2sin(1\u03c0(4(-1x^2 + 5x + 2) - 2) + 2) - 2) - 1| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "259", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2|(-4(4(1sin(3\u03c0x - 1) - 3) - 4)^2 + 5) - 2| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(-3cos(2\u03c0(4(1|x - 1| + 1) + 1) - 2) + 1) + 1)/(2(-3cos(2\u03c0(4(1|x - 1| + 1) + 1) - 2) + 1)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1((-3(-5(-1sin(2\u03c0x + 3) - 3) + 2) + 2)/(2(-5(-1sin(2\u03c0x + 3) - 3) + 2) + 3))^2 + 2((-3(-5(-1sin(2\u03c0x + 3) - 3) + 2) + 2)/(2(-5(-1sin(2\u03c0x + 3) - 3) + 2) + 3)) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1cos(1\u03c0(-5(3((-2x)/(2x + 3)) - 1)^2 + 1(3((-2x)/(2x + 3)) - 1) - 4) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "113", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-2(2|(1cos(1\u03c0x + 2) - 1) + 3| + 3) + 1)/(1(2|(1cos(1\u03c0x + 2) - 1) + 3| + 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(3\u03c0(-1|(1x) - 2| + 3) - 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(1\u03c0(-1((-2(1|x - 2| + 2) - 1)/(-1(1|x - 2| + 2) + 3)) + 3) - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2cos(2\u03c0(4(1|(-5x + 1) + 2| + 2)^2 + 1(1|(-5x + 1) + 2| + 2) - 5) - 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "199", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(3cos(2\u03c0((3(2|x + 3| - 1) - 3)/(-2(2|x + 3| - 1))) + 2) - 1)^2 - 5(3cos(2\u03c0((3(2|x + 3| - 1) - 3)/(-2(2|x + 3| - 1))) + 2) - 1) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 5(-3cos(1\u03c0(-1|x + 3| - 3) + 3)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|(-2e^(-1(1x^2 - 3x + 2)) + 2)| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -5(1cos(2\u03c0((-3(3x^2 + 2x + 2) - 3)/(1(3x^2 + 2x + 2) - 2)) + 3) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1(3(-1cos(2\u03c0(-2|x - 3| + 3) - 1) + 3) - 4)^2 + 2(3(-1cos(2\u03c0(-2|x - 3| + 3) - 1) + 3) - 4) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(2|(2cos(1\u03c0((2x - 1)/(-3x + 2)) + 1) + 2) + 3| + 1) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1(-1e^(-1(4(2|x - 1| + 3) + 5)) + 3)^2 - 1(-1e^(-1(4(2|x - 1| + 3) + 5)) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(-2cos(1\u03c0(-4x^2 + 4x - 5) - 2) + 2) - 3)/(-1(-2cos(1\u03c0(-4x^2 + 4x - 5) - 2) + 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "569", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2sin(1\u03c0(1(-4x^2 + 5x + 5) - 3) + 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "283", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (3(-3cos(3\u03c0(5x^2 + 2x + 4) + 2) + 3) - 2)/(-2(-3cos(3\u03c0(5x^2 + 2x + 4) + 2) + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "500", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (1(1|(1sin(2\u03c0x - 1) - 3) + 1| - 2) - 2)/(-3(1|(1sin(2\u03c0x - 1) - 3) + 1| - 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (3(-1(-3sin(1\u03c0(1|x| + 2) - 2) + 2)^2 + 3(-3sin(1\u03c0(1|x| + 2) - 2) + 2) - 4) + 1)/(-1(-1(-3sin(1\u03c0(1|x| + 2) - 2) + 2)^2 + 3(-3sin(1\u03c0(1|x| + 2) - 2) + 2) - 4) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(2((1x - 1)/(-2x - 1)) - 4) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-3(2sin(1\u03c0(1(2|x - 3| + 3) + 2) + 2) - 1) + 1)/(-2(2sin(1\u03c0(1(2|x - 3| + 3) + 2) + 2) - 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2(-2cos(1\u03c0(1|x - 2| - 2) - 1) - 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(3cos(2\u03c0(-5(-2x^2 - 3x - 5) + 5) - 3)) - 1)/(2(3cos(2\u03c0(-5(-2x^2 - 3x - 5) + 5) - 3)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "234", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1sin(2\u03c0(5(1|x - 2| + 3)^2 - 1(1|x - 2| + 3) + 4) - 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "470", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(3cos(3\u03c0(5x + 5) + 1))^2 + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2(-1cos(2\u03c0(2|x - 2|) + 2) - 3)^2 - 5(-1cos(2\u03c0(2|x - 2|) + 2) - 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 5(3sin(2\u03c0(-3|((-1x - 3)/(1x - 2))| + 1) + 1) - 3) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3sin(3\u03c0(-5(5x^2 - 3x - 2) + 4) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "227", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2(-1cos(1\u03c0(-2((2x)/(-3x - 1)) - 1) - 1))^2 + 3(-1cos(1\u03c0(-2((2x)/(-3x - 1)) - 1) - 1)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1((2(3|(-1cos(3\u03c0x - 2) - 3) - 1| + 2) + 3)/(-1(3|(-1cos(3\u03c0x - 2) - 3) - 1| + 2) + 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 5(-3(1cos(2\u03c0(-3|x| - 2) - 1) + 2)^2 - 2(1cos(2\u03c0(-3|x| - 2) - 1) + 2) - 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3(3|(-3cos(3\u03c0x + 2)) + 1| + 3)^2 - 4(3|(-3cos(3\u03c0x + 2)) + 1| + 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (2(-2sin(3\u03c0(-1(-2|x - 2|)^2 + 3(-2|x - 2|) - 2) - 3) + 1) - 2)/(-3(-2sin(3\u03c0(-1(-2|x - 2|)^2 + 3(-2|x - 2|) - 2) - 3) + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "148", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3(-4(3sin(2\u03c0x - 3) + 3) - 2) + 3)/(-2(-4(3sin(2\u03c0x - 3) + 3) - 2) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1((-1(1cos(1\u03c0x + 1) - 1) + 2)/(2(1cos(1\u03c0x + 1) - 1) + 1)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1|((3(-3cos(3\u03c0x - 3) - 3))/(-2(-3cos(3\u03c0x - 3) - 3) + 1)) - 1| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3(3((-3(-2cos(2\u03c0x - 1) + 2) - 3)/(1(-2cos(2\u03c0x - 1) + 2) + 2)) - 1)^2 + 5(3((-3(-2cos(2\u03c0x - 1) + 2) - 3)/(1(-2cos(2\u03c0x - 1) + 2) + 2)) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2((3(3e^(-2(2x - 4)) - 1) + 3)/(-1(3e^(-2(2x - 4)) - 1) + 3))^2 + 1((3(3e^(-2(2x - 4)) - 1) + 3)/(-1(3e^(-2(2x - 4)) - 1) + 3)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3((-1(2cos(3\u03c0x) - 1) - 1)/(2(2cos(3\u03c0x) - 1) - 3))^2 + 5((-1(2cos(3\u03c0x) - 1) - 1)/(2(2cos(3\u03c0x) - 1) - 3)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1(-1|(-5(-2cos(2\u03c0x + 3))^2 - 2(-2cos(2\u03c0x + 3)) - 5) + 1| + 2) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(2cos(2\u03c0(-2(-3x + 1)^2 - 1(-3x + 1)) - 2) - 1) - 1)/(1(2cos(2\u03c0(-2(-3x + 1)^2 - 1(-3x + 1)) - 2) - 1)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1sin(1\u03c0(2|(2x^2 - 5) - 1|)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3sin(1\u03c0(-4(-5x^2 + 1x + 4)) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "253", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2sin(1\u03c0(-5(-5x + 1)^2 - 2) - 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "424", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 4((2(2cos(1\u03c0x)) - 2)/(3(2cos(1\u03c0x)) + 1))^2 + 2((2(2cos(1\u03c0x)) - 2)/(3(2cos(1\u03c0x)) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(-3e^(1(-3x + 1)) - 1)^2 - 2(-3e^(1(-3x + 1)) - 1) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1((3(1cos(2\u03c0(1x^2 + 5x - 2) + 2) - 2) + 3)/(1(1cos(2\u03c0(1x^2 + 5x - 2) + 2) - 2) - 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "424", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1sin(2\u03c0(-1(-5x + 2)^2 + 5(-5x + 2) + 2) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "234", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2|(-1sin(3\u03c0(-1x + 5) - 2) + 1)| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(2\u03c0(-2|(-3x + 4)| - 1) + 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-2(3|(-2(-2cos(2\u03c0x + 3) + 2) + 2) - 2| - 2) - 3)/(1(3|(-2(-2cos(2\u03c0x + 3) + 2) + 2) - 2| - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (2(2sin(3\u03c0(2(-3|x| + 3)^2 - 1(-3|x| + 3) - 1) - 2)) - 2)/(-3(2sin(3\u03c0(2(-3|x| + 3)^2 - 1(-3|x| + 3) - 1) - 2)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "500", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3(1|(1cos(2\u03c0((2x + 1)/(1x + 3))) - 2) + 3| - 3)^2 - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(2\u03c0(-3|(-1x + 4) - 1| + 1) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (1(-2cos(2\u03c0(-3|x| + 3)) - 3) - 3)/(1(-2cos(2\u03c0(-3|x| + 3)) - 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3sin(2\u03c0(-5(3(1|x - 1| + 3)^2 - 5(1|x - 1| + 3) + 1) - 5)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "250", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2(2|(1cos(1\u03c0((-2x)/(-1x - 2)) - 2) - 1)| + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(3(2|(-2cos(1\u03c0x + 1) + 3) + 1| - 3)) + 1)/(1(3(2|(-2cos(1\u03c0x + 1) + 3) + 1| - 3)) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3cos(1\u03c0(2|(-1x^2 - 2x - 3) - 1|)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "204", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(3|(3cos(2\u03c0x) + 1) - 2| - 2) - 1)/(3(3|(3cos(2\u03c0x) + 1) - 2| - 2) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1(2cos(1\u03c0(-2|x - 2| - 1)) + 2)^2 + 1(2cos(1\u03c0(-2|x - 2| - 1)) + 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3sin(1\u03c0((3(-4x^2 + 4x + 4) - 3)/(1(-4x^2 + 4x + 4) + 2)) - 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1((-2(2cos(1\u03c0(-2|x| + 1) + 2)) + 1)/(-3(2cos(1\u03c0(-2|x| + 1) + 2)) - 2))^2 + 2((-2(2cos(1\u03c0(-2|x| + 1) + 2)) + 1)/(-3(2cos(1\u03c0(-2|x| + 1) + 2)) - 2)) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(-2sin(2\u03c0((-3x + 1)/(1x))) + 3)^2 - 2(-2sin(2\u03c0((-3x + 1)/(1x))) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(3cos(2\u03c0(2x^2 + 3x - 2) + 2) - 1) - 2)/(2(3cos(2\u03c0(2x^2 + 3x - 2) + 2) - 1) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "395", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2|(-4(3sin(1\u03c0x + 2) + 3)^2 - 1(3sin(1\u03c0x + 2) + 3) + 5)| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3(-3(-3cos(1\u03c0x + 2) + 2)^2 - 3(-3cos(1\u03c0x + 2) + 2) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3|(5(-3sin(2\u03c0x + 3) - 1)^2 + 5(-3sin(2\u03c0x + 3) - 1) + 1)| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(1|(-4(-2sin(2\u03c0x + 1) - 1)^2 - 2(-2sin(2\u03c0x + 1) - 1) - 5)| + 2) - 3)/(-1(1|(-4(-2sin(2\u03c0x + 1) - 1)^2 - 2(-2sin(2\u03c0x + 1) - 1) - 5)| + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(-1(-2sin(1\u03c0x) - 3) - 1) - 3)/(3(-1(-2sin(1\u03c0x) - 3) - 1) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-3(-5(-3(1cos(3\u03c0x + 1) + 3)^2 + 3(1cos(3\u03c0x + 1) + 3) - 3) + 5) - 3)/(-3(-5(-3(1cos(3\u03c0x + 1) + 3)^2 + 3(1cos(3\u03c0x + 1) + 3) - 3) + 5) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(1cos(2\u03c0(1x + 1)) + 1)^2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1sin(1\u03c0((3(2(4x + 4)^2 - 4) - 1)/(-2(2(4x + 4)^2 - 4))) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4((-1(-3sin(3\u03c0x)))/(-3(-3sin(3\u03c0x)) - 2))^2 + 5((-1(-3sin(3\u03c0x)))/(-3(-3sin(3\u03c0x)) - 2)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2cos(2\u03c0(4(2|(-5x + 2) + 3| - 2)^2 - 3) - 1) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "260", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3(-3|(3(1sin(2\u03c0x - 2) + 1) + 5) + 1| + 3)^2 - 4(-3|(3(1sin(2\u03c0x - 2) + 1) + 5) + 1| + 3) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2sin(1\u03c0(-4(-1((-3x + 3)/(1x))^2 + 1((-3x + 3)/(1x)) + 2) + 1) + 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-2(-5(1sin(1\u03c0(4x + 5) - 1) - 1)^2 + 1(1sin(1\u03c0(4x + 5) - 1) - 1) + 5) - 3)/(-2(-5(1sin(1\u03c0(4x + 5) - 1) - 1)^2 + 1(1sin(1\u03c0(4x + 5) - 1) - 1) + 5) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3cos(1\u03c0(1((-1(2x + 3) + 3)/(1(2x + 3) - 2))^2 + 5((-1(2x + 3) + 3)/(1(2x + 3) - 2)) - 1) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3|(1e^(2(1x - 5)) - 3) - 3| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1|((1(-1cos(2\u03c0(-5x + 1) + 3) - 1) + 2)/(3(-1cos(2\u03c0(-5x + 1) + 3) - 1) - 2)) - 1| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(2\u03c0(-1|((1x + 3)/(1x)) + 3| - 1) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(-2(-2cos(2\u03c0x + 2)) - 4)^2 + 2(-2(-2cos(2\u03c0x + 2)) - 4) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1(3(-3cos(3\u03c0(-3|x - 1| - 3) - 2)) + 4)^2 - 5(3(-3cos(3\u03c0(-3|x - 1| - 3) - 2)) + 4) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(-5(1(1cos(1\u03c0x + 2) + 1)^2 - 3(1cos(1\u03c0x + 2) + 1) - 5) + 2) - 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1((-2(3sin(1\u03c0x - 1) - 3) + 3)/(2(3sin(1\u03c0x - 1) - 3) + 2))^2 + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(2\u03c0(4((-2(2|x|) + 3)/(2(2|x|) + 2)) + 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(2\u03c0((1(-4(-2|x - 1| + 3) + 4) + 3)/(3(-4(-2|x - 1| + 3) + 4) + 2)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(-1|(2cos(1\u03c0x) + 1) + 3| - 3))/(1(-1|(2cos(1\u03c0x) + 1) + 3| - 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -4(1(2sin(2\u03c0(3|x - 1| + 3) + 2) - 1)^2 + 4(2sin(2\u03c0(3|x - 1| + 3) + 2) - 1) - 5) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1cos(2\u03c0(-2(3x - 5)^2 + 1(3x - 5) + 4) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "246", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2sin(3\u03c0(-1(-1|(-5x + 4) - 1| - 1)^2 + 3(-1|(-5x + 4) - 1| - 1) - 3) + 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "228", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1cos(2\u03c0(3|(4x + 4) - 1| - 3) + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "239", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(-1|(-2(2sin(2\u03c0x - 3) - 2)^2 + 4(2sin(2\u03c0x - 3) - 2) - 5) - 3| - 1) + 1)/(1(-1|(-2(2sin(2\u03c0x - 3) - 2)^2 + 4(2sin(2\u03c0x - 3) - 2) - 5) - 3| - 1) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1((1(-3sin(3\u03c0x + 2)) + 2)/(2(-3sin(3\u03c0x + 2)) + 3)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(3\u03c0(-3((2(2|x - 2|) - 2)/(-1(2|x - 2|) + 3)) + 1) - 3) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(-3(-1e^(1(2x + 2)) + 1)^2 - 2(-1e^(1(2x + 2)) + 1) - 1) + 1)/(-1(-3(-1e^(1(2x + 2)) + 1)^2 - 2(-1e^(1(2x + 2)) + 1) - 1) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(2cos(3\u03c0((3(-1x + 4) - 1)/(3(-1x + 4) + 2)) + 2) - 3)^2 - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(-5(1cos(3\u03c0x - 3))^2 - 1(1cos(3\u03c0x - 3)) - 4) - 1)/(-2(-5(1cos(3\u03c0x - 3))^2 - 1(1cos(3\u03c0x - 3)) - 4) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2((1(-2cos(2\u03c0x + 2) + 1) + 1)/(-1(-2cos(2\u03c0x + 2) + 1) - 2))^2 - 1((1(-2cos(2\u03c0x + 2) + 1) + 1)/(-1(-2cos(2\u03c0x + 2) + 1) - 2)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3|(-4(-2cos(1\u03c0((-2x + 2)/(-3x))) - 2)^2 + 3(-2cos(1\u03c0((-2x + 2)/(-3x))) - 2) - 1) + 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1cos(3\u03c0(-1|((1(-1x - 4) - 2)/(-3(-1x - 4) - 3)) - 1| + 1) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(-4(-3sin(1\u03c0x + 2) + 3)) - 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1sin(2\u03c0(-2((-3x - 1)/(-1x + 2))^2 - 5) + 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5((-1(2cos(3\u03c0(4x) - 1)))/(2(2cos(3\u03c0(4x) - 1)) + 1))^2 - 2((-1(2cos(3\u03c0(4x) - 1)))/(2(2cos(3\u03c0(4x) - 1)) + 1)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "503", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1sin(3\u03c0(1|(-1((-1x - 1)/(3x + 2)) + 4) - 2| + 3) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(-3sin(3\u03c0(-5(-2|x - 3| + 2)^2 + 1(-2|x - 3| + 2) - 5) + 1) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "229", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(1|(2x + 5)| + 3) - 3)/(2(1|(2x + 5)| + 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(2cos(1\u03c0(-3|(5x - 3) + 1| - 2) + 2) - 1)^2 - 2(2cos(1\u03c0(-3|(5x - 3) + 1| - 2) + 2) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "579", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1sin(1\u03c0(3|(1x - 2) + 1| - 3) + 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1(3cos(3\u03c0(-1(-2|x - 3|)^2 + 3) - 2) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "251", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4(3|(1sin(2\u03c0x + 2) - 1) - 3|) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3(1sin(2\u03c0((-2x + 1)/(3x + 3)) - 2) - 2)^2 + 5(1sin(2\u03c0((-2x + 1)/(3x + 3)) - 2) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2cos(2\u03c0((3(3|x - 3| + 2) + 2)/(-2(3|x - 3| + 2) + 3)) - 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 4((2(-1cos(2\u03c0x + 1)) - 3)/(-3(-1cos(2\u03c0x + 1)) - 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2|(2cos(1\u03c0((3x + 2)/(-3x + 4)) - 1)) - 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(1\u03c0(-2(2|x - 1| + 1)^2 - 5(2|x - 1| + 1) - 4) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "275", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-2(-1(1sin(2\u03c0x - 1) - 3) - 3) - 1)/(3(-1(1sin(2\u03c0x - 1) - 3) - 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3(-3sin(3\u03c0((-2x + 3)/(-3x + 2)) + 2) + 3)^2 + 2(-3sin(3\u03c0((-2x + 3)/(-3x + 2)) + 2) + 3) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(2|(2cos(1\u03c0x + 1) + 1) - 2|) + 3)/(-2(2|(2cos(1\u03c0x + 1) + 1) - 2|) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(1\u03c0(5(-1|(5x - 1) - 2| - 2)^2 - 4(-1|(5x - 1) - 2| - 2) - 2) + 3) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "442", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4(-2cos(1\u03c0(-2|((-1x + 2)/(2x - 3))|) - 1) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2(-3cos(1\u03c0((-1x + 2)/(3x - 2)) + 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(-4(-1cos(1\u03c0x) - 3)^2 - 3(-1cos(1\u03c0x) - 3) - 1) - 2)/(-3(-4(-1cos(1\u03c0x) - 3)^2 - 3(-1cos(1\u03c0x) - 3) - 1) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(2\u03c0((-3(-5x - 1) - 2)/(-2(-5x - 1) - 2)) - 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3(-1cos(3\u03c0((3x - 1)/(-3x + 4)) + 3) - 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2sin(2\u03c0((2(-1x^2 - 4x - 1) - 3)/(2(-1x^2 - 4x - 1) + 1)) + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(-1cos(2\u03c0(3|((-2x + 3)/(-3x + 1)) - 2| + 2) - 3) + 1)^2 - 3(-1cos(2\u03c0(3|((-2x + 3)/(-3x + 1)) - 2| + 2) - 3) + 1) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -4(-3(2cos(3\u03c0x + 3) + 1)^2 - 5(2cos(3\u03c0x + 3) + 1) - 5) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(-1|(2e^(1(-2x^2 + 4x + 3)) + 1) - 1| - 2) + 1)/(1(-1|(2e^(1(-2x^2 + 4x + 3)) + 1) - 1| - 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5(-2sin(2\u03c0(-1|x - 2| + 3) - 3) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3cos(1\u03c0(-1(2x)^2) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(4(2sin(3\u03c0(2x^2 + 4x - 4)) + 3) - 1))/(3(4(2sin(3\u03c0(2x^2 + 4x - 4)) + 3) - 1) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "463", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -5(-3cos(3\u03c0((-2(2x^2 - 4x - 3) + 3)/(3(2x^2 - 4x - 3) + 1))) + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 4((-2(-5(-3cos(2\u03c0x - 3) - 2)^2 + 3(-3cos(2\u03c0x - 3) - 2) - 1) - 2)/(-3(-5(-3cos(2\u03c0x - 3) - 2)^2 + 3(-3cos(2\u03c0x - 3) - 2) - 1) + 4)) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3(-1(-3cos(3\u03c0(-3|x + 3| + 3) + 1) - 3)^2 - 1(-3cos(3\u03c0(-3|x + 3| + 3) + 1) - 3) + 1) + 3)/(2(-1(-3cos(3\u03c0(-3|x + 3| + 3) + 1) - 3)^2 - 1(-3cos(3\u03c0(-3|x + 3| + 3) + 1) - 3) + 1) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(3\u03c0((-3(5x + 1) - 3)/(-1(5x + 1))) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1sin(2\u03c0(1((-2x)/(3x + 2))^2 + 5((-2x)/(3x + 2))) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4(-2sin(1\u03c0((1x + 2)/(3x - 2)) + 2) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1cos(2\u03c0(2(-1((3x + 1)/(-2x + 3)))^2 + 5(-1((3x + 1)/(-2x + 3))) + 5) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(-2cos(1\u03c0(-4(3|x - 1| + 1) + 5) + 1)) + 1)/(-3(-2cos(1\u03c0(-4(3|x - 1| + 1) + 5) + 1)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1((3(-4(-3cos(1\u03c0x) + 3)^2 + 4(-3cos(1\u03c0x) + 3) - 3) - 3)/(1(-4(-3cos(1\u03c0x) + 3)^2 + 4(-3cos(1\u03c0x) + 3) - 3) - 2)) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3|(2(2sin(3\u03c0((-2x + 1)/(-2x - 1)) - 1) + 2)^2 + 3(2sin(3\u03c0((-2x + 1)/(-2x - 1)) - 1) + 2) + 3) - 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(2sin(2\u03c0(-2|(-5x + 3) + 2| + 3) + 2) - 1)^2 + 4(2sin(2\u03c0(-2|(-5x + 3) + 2| + 3) + 2) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "799", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3|((2(-3(1sin(3\u03c0x - 1) + 1)^2 + 3(1sin(3\u03c0x - 1) + 1) + 2) - 3)/(2(-3(1sin(3\u03c0x - 1) + 1)^2 + 3(1sin(3\u03c0x - 1) + 1) + 2) + 2))| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1cos(3\u03c0(-2|((-2x - 1)/(2x + 2)) - 1| + 1) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(1sin(2\u03c0(1|(2x^2 + 1x - 5)| - 2) - 3) + 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "577", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3(-2|(-2cos(3\u03c0(-5x^2 - 1x - 5) - 2)) + 2|) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "486", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(1\u03c0((3(-2(-1x + 1)^2 - 2(-1x + 1) + 1) - 1)/(3(-2(-1x + 1)^2 - 2(-1x + 1) + 1) + 1)) + 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1|(-2(2cos(1\u03c0x) + 3) - 4) - 1| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2cos(2\u03c0(-1(-2x^2 + 4x - 3)) - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(1(-2cos(3\u03c0(3|x + 1|) - 1)) - 1)^2 + 5(1(-2cos(3\u03c0(3|x + 1|) - 1)) - 1) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "361", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3cos(1\u03c0(-2(2|(1x + 5) - 1|)^2 + 5(2|(1x + 5) - 1|) - 5) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "236", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2(-2sin(2\u03c0((-2x - 1)/(3x + 1)) + 3) - 2)^2 + 5(-2sin(2\u03c0((-2x - 1)/(3x + 1)) + 3) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (3(-1(4(3cos(3\u03c0x + 2) - 3) - 2)^2 + 4(4(3cos(3\u03c0x + 2) - 3) - 2)) + 1)/(1(-1(4(3cos(3\u03c0x + 2) - 3) - 2)^2 + 4(4(3cos(3\u03c0x + 2) - 3) - 2)) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(3\u03c0(4(3x - 1)^2 + 3(3x - 1) + 1) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(-3sin(3\u03c0(-1|(5x + 2) - 3| - 2) - 2)) + 2)/(-1(-3sin(3\u03c0(-1|(5x + 2) - 3| - 2) - 2)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "301", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1(-5(2|(-3cos(3\u03c0x + 2) - 1) - 2| - 2)^2 - 2(2|(-3cos(3\u03c0x + 2) - 1) - 2| - 2) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3|((1(3cos(2\u03c0x - 1) - 3) - 3)/(-3(3cos(2\u03c0x - 1) - 3) - 2)) - 2| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1(-3sin(3\u03c0(-3x + 4) + 3) - 3)^2 + 1(-3sin(3\u03c0(-3x + 4) + 3) - 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2cos(3\u03c0(-5(3|x - 3| - 2) - 2)) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "467", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(3|(5(1cos(2\u03c0x) + 1)^2 - 3(1cos(2\u03c0x) + 1) + 2) + 3|) - 3)/(-3(3|(5(1cos(2\u03c0x) + 1)^2 - 3(1cos(2\u03c0x) + 1) + 2) + 3|) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1|(1cos(2\u03c0((1(4x + 1) - 1)/(-1(4x + 1))))) + 1| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(2\u03c0(-1(2|(4x^2 + 3x + 2) + 2| - 1) - 1) - 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "244", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4((3(-2|(2cos(2\u03c0x - 3) - 2) - 2|) - 3)/(-2(-2|(2cos(2\u03c0x - 3) - 2) - 2|) + 3))^2 + 2((3(-2|(2cos(2\u03c0x - 3) - 2) - 2|) - 3)/(-2(-2|(2cos(2\u03c0x - 3) - 2) - 2|) + 3)) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1sin(2\u03c0(1|(4((-1x + 2)/(-2x - 3)) - 4) - 1| + 2) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(3\u03c0(-2(-1|((-2x - 1)/(2x - 1))| - 3) - 3) - 1) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1cos(2\u03c0(2(1x^2 - 2x + 2) + 1) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "565", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(5(-2|(3cos(1\u03c0x - 3)) + 3| - 3)^2 - 1(-2|(3cos(1\u03c0x - 3)) + 3| - 3) + 3) - 2)/(-2(5(-2|(3cos(1\u03c0x - 3)) + 3| - 3)^2 - 1(-2|(3cos(1\u03c0x - 3)) + 3| - 3) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(2\u03c0((-2(-5x^2 + 4x + 5) + 2)/(3(-5x^2 + 4x + 5))) + 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1sin(1\u03c0(-2(4(-3|x - 1|)^2 - 5(-3|x - 1|) - 2) + 4) + 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "472", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0(-4((-1x + 1)/(1x)) + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(3\u03c0(-5(1|x - 3| + 3) + 4) - 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "299", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 5(3cos(1\u03c0(-5x + 3) + 2) + 3)^2 + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1(5(-2cos(2\u03c0x) + 2) + 4)^2 - 2(5(-2cos(2\u03c0x) + 2) + 4) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1|(-1cos(3\u03c0(4x^2 + 3x - 5)) + 2) + 2| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "243", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1((-2(1cos(1\u03c0x + 3) + 2) + 1)/(-2(1cos(1\u03c0x + 3) + 2))) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(2\u03c0((-3(3|(4x^2 - 5x + 3) - 3|) + 2)/(3(3|(4x^2 - 5x + 3) - 3|) - 1)) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4(1cos(1\u03c0(2((-2x + 3)/(2x))^2 - 3((-2x + 3)/(2x)) - 1) + 1) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(-2|(3sin(3\u03c0x) - 2) + 2| - 1)^2 - 2(-2|(3sin(3\u03c0x) - 2) + 2| - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(-1e^(-2(4x - 5)) + 3) - 3)/(-1(-1e^(-2(4x - 5)) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3(3cos(2\u03c0(2((2x - 3)/(-3x + 1)) + 3) + 1))^2 - 1(3cos(2\u03c0(2((2x - 3)/(-3x + 1)) + 3) + 1)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(2\u03c0((2(1x - 3))/(2(1x - 3) + 2)) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(-2|(1cos(3\u03c0x - 2) - 2) + 3|)^2 - 3(-2|(1cos(3\u03c0x - 2) - 2) + 3|) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(3cos(3\u03c0((-3x)/(-2x - 3))) - 1) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2sin(1\u03c0(-3|(-1x^2 - 2x - 3) + 3| - 1) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "277", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(-2cos(2\u03c0((-3(4x + 2) - 3)/(-2(4x + 2) - 3)) + 2) - 1) + 2| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(5(2cos(3\u03c0x + 2) - 1) + 3) + 1)/(-3(5(2cos(3\u03c0x + 2) - 1) + 3) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3|(-3(-1cos(1\u03c0((2x + 2)/(-2x)) - 2) - 2) - 5) + 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(-1|(1cos(3\u03c0x + 1) + 2) - 1| - 2) + 2)/(-2(-1|(1cos(3\u03c0x + 1) + 2) - 1| - 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 4(1cos(3\u03c0((-1(1x + 5))/(2(1x + 5) - 1))))^2 - 1(1cos(3\u03c0((-1(1x + 5))/(2(1x + 5) - 1)))) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 5(3|(3cos(1\u03c0((2x - 2)/(2x - 1)) - 2))|)^2 + 4(3|(3cos(1\u03c0((2x - 2)/(2x - 1)) - 2))|) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1sin(1\u03c0(-4(4x^2 + 2x - 1) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "239", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3sin(1\u03c0(-3|(5x - 5)| - 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2cos(1\u03c0(-2|(-3x^2 - 1x + 4)| - 1)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "227", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (1(-3cos(3\u03c0(1x - 4) + 2) + 1) - 2)/(3(-3cos(3\u03c0(1x - 4) + 2) + 1)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4(-5((2(-1cos(1\u03c0x - 2) + 2) - 2)/(-2(-1cos(1\u03c0x - 2) + 2) + 1))^2 - 1((2(-1cos(1\u03c0x - 2) + 2) - 2)/(-2(-1cos(1\u03c0x - 2) + 2) + 1)) + 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3cos(2\u03c0(2|((-1x + 2)/(-1x - 2)) - 2|) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-2|(2e^(-2(-5x + 2)) + 1) - 3|))/(1(-2|(2e^(-2(-5x + 2)) + 1) - 3|) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-1sin(1\u03c0(2|x + 2|) - 3) + 3) - 3)/(-1(-1sin(1\u03c0(2|x + 2|) - 3) + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1|(-1(2sin(1\u03c0(3x^2 + 1x - 1) + 3) - 3)) - 2| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "484", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2(1cos(3\u03c0((-3(4x + 3) + 3)/(-2(4x + 3) + 3)) - 2) + 3)^2 - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(1|(3cos(3\u03c0x - 1) - 2) - 2|)^2 - 1(1|(3cos(3\u03c0x - 1) - 2) - 2|) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|(2(-1sin(3\u03c0((-1x - 2)/(2x - 3))) + 1) - 5) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3sin(2\u03c0(2(-3|((-2x - 2)/(1x)) + 2| + 2)^2 - 1(-3|((-2x - 2)/(1x)) + 2| + 2))) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(2\u03c0(3((1x)/(-2x + 3)) - 5) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2|(2cos(2\u03c0((-1(-5x) + 2)/(-1(-5x) - 3)) + 3)) - 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3(-1sin(3\u03c0(-1(-3|x| + 1) - 2) - 1) + 1))/(-2(-1sin(3\u03c0(-1(-3|x| + 1) - 2) - 1) + 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-1(-1(2|(1cos(3\u03c0x + 3) + 1) + 1| - 3) - 1) - 2)/(2(-1(2|(1cos(3\u03c0x + 3) + 1) + 1| - 3) - 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -5(1sin(3\u03c0(2(-1|x - 3| + 2) - 3) + 3) - 3)^2 + 5(1sin(3\u03c0(2(-1|x - 3| + 2) - 3) + 3) - 3) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|((-2(2cos(2\u03c0x + 3) + 1) + 2)/(-3(2cos(2\u03c0x + 3) + 1) - 3)) + 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2(2sin(2\u03c0(-3|x + 3| - 2) + 3) - 1)^2 - 2(2sin(2\u03c0(-3|x + 3| - 2) + 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "239", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3cos(2\u03c0(5(3|x + 3| + 2)^2 - 2(3|x + 3| + 2) - 5) + 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "209", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3sin(3\u03c0(2|(-1x^2 + 4x - 4) + 1|) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "247", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3|(-2(2cos(2\u03c0(-4x + 3) - 2) - 3)^2 - 2(2cos(2\u03c0(-4x + 3) - 2) - 3) - 2) - 2| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (2(-1|(3sin(1\u03c0x - 3) - 3)| + 3) + 2)/(2(-1|(3sin(1\u03c0x - 3) - 3)| + 3) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1cos(2\u03c0(-2|(-2x - 1) - 1| + 2) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "161", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1sin(3\u03c0((1(-2|(-5x^2 - 1x) + 1| + 3) - 3)/(2(-2|(-5x^2 - 1x) + 1| + 3) - 1))) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -4((-1(-5(1e^(-2x) + 2)^2 - 5(1e^(-2x) + 2) - 5) - 3)/(-3(-5(1e^(-2x) + 2)^2 - 5(1e^(-2x) + 2) - 5) + 1)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3|(4(-2sin(2\u03c0x + 3) - 1) + 5)| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(-1|(2(-3sin(3\u03c0x - 2) + 3)^2 - 1(-3sin(3\u03c0x - 2) + 3) - 1) + 2| + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-3cos(2\u03c0(1(2|x - 1| - 3) - 5) - 2) + 2) - 3)/(-3(-3cos(2\u03c0(1(2|x - 1| - 3) - 5) - 2) + 2) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(1cos(3\u03c0(-5x + 4) + 1) + 3) + 2)/(3(1cos(3\u03c0(-5x + 4) + 1) + 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "150", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (1(-4(2cos(2\u03c0(3x^2 - 5x - 2) + 3) - 3) + 1) - 2)/(2(-4(2cos(2\u03c0(3x^2 - 5x - 2) + 3) - 3) + 1) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "117", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(1|(2sin(3\u03c0x - 3) - 3) - 3| - 2) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(3\u03c0(5(3|x| - 3)^2 + 2(3|x| - 3) + 3) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "229", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1((-3(-2cos(1\u03c0(2|x + 1| - 3) + 2) + 1) - 2)/(1(-2cos(1\u03c0(2|x + 1| - 3) + 2) + 1) - 3)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|((-3(-3sin(1\u03c0x + 1) + 1) - 1)/(1(-3sin(1\u03c0x + 1) + 1) + 1))| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(1\u03c0(2|(2x^2 - 4x) - 2| + 2) - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "288", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1(3(1sin(3\u03c0x - 2) - 1)^2 - 3(1sin(3\u03c0x - 2) - 1) + 4) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3(3cos(3\u03c0((-3x + 3)/(-1x - 3)) - 3) - 1) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(1\u03c0(4((1x + 1)/(3x - 2))^2 + 1((1x + 1)/(3x - 2)) + 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(3\u03c0(-1(-1|x - 2| - 1)^2 + 2(-1|x - 2| - 1) + 5) + 2) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "316", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3cos(1\u03c0(3|((2(2x + 2))/(-2(2x + 2) - 2)) - 1| - 3) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2sin(1\u03c0((3(3|(1x^2 - 4x - 3) + 1| - 1) - 1)/(-2(3|(1x^2 - 4x - 3) + 1| - 1) - 1)) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2cos(2\u03c0(-1|(-1x - 2) - 3|) + 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(-3sin(3\u03c0(-3x^2 - 2x + 1) + 1)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "526", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3cos(3\u03c0(1(2|x - 2|) - 3) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1sin(1\u03c0(4(1|(-5x + 1)| - 3)^2 - 3(1|(-5x + 1)| - 3) + 2)) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "210", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3(2((-3(-1sin(1\u03c0x + 3)) + 3)/(-3(-1sin(1\u03c0x + 3)) + 2)) + 2)^2 + 3(2((-3(-1sin(1\u03c0x + 3)) + 3)/(-3(-1sin(1\u03c0x + 3)) + 2)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(2(5(3cos(3\u03c0x + 3) - 1) + 2)^2 - 1(5(3cos(3\u03c0x + 3) - 1) + 2) + 2) - 2| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (1(2|(-2sin(1\u03c0x + 2) - 1) + 3| + 2) + 3)/(-1(2|(-2sin(1\u03c0x + 2) - 1) + 3| + 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(3(1|((2x + 2)/(-3x - 3)) + 1|)^2 - 5(1|((2x + 2)/(-3x - 3)) + 1|) - 1) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2(-3|(3cos(3\u03c0x - 3) - 1) + 3| + 3)^2 + 3(-3|(3cos(3\u03c0x - 3) - 1) + 3| + 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3(-2sin(2\u03c0(-1|(-3x - 1) + 2| - 1) - 3) - 2)^2 + 2(-2sin(2\u03c0(-1|(-3x - 1) + 2| - 1) - 3) - 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(-1sin(2\u03c0(-5x^2 + 1x + 1) - 2) - 1) - 2| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "247", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-1(-4(-1cos(1\u03c0(-1x^2 + 1x + 3) + 3) - 3) + 4) - 2)/(3(-4(-1cos(1\u03c0(-1x^2 + 1x + 3) + 3) - 3) + 4) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "203", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2cos(1\u03c0(3((-1x - 2)/(3x - 2))^2 - 5((-1x - 2)/(3x - 2)) + 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2|(2cos(1\u03c0(1(1x - 3)^2 + 4(1x - 3)) + 3) - 1)| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "391", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1|(1sin(2\u03c0((-3x - 3)/(1x)) - 3) + 3) + 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(3|(-2sin(3\u03c0(4x^2 - 1x - 5) + 3) + 2) + 3| + 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "485", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(-2cos(3\u03c0((2(-1x + 5))/(-2(-1x + 5) - 2))) + 2)^2 - 1(-2cos(3\u03c0((2(-1x + 5))/(-2(-1x + 5) - 2))) + 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3(1sin(3\u03c0((-3x)/(-1x + 2)) - 2) + 2)^2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1(-3sin(3\u03c0((2x - 2)/(2x - 3)) - 1) + 2) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-2(3cos(2\u03c0(-1(-2|x - 3| - 2)^2 + 2(-2|x - 3| - 2) - 2)) + 3) + 3)/(1(3cos(2\u03c0(-1(-2|x - 3| - 2)^2 + 2(-2|x - 3| - 2) - 2)) + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "365", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2|((3(-1e^(1(-5x^2 - 1x - 1)) + 3))/(1(-1e^(1(-5x^2 - 1x - 1)) + 3)))| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1cos(2\u03c0(-1((2(3|x - 2| + 1) + 2)/(-2(3|x - 2| + 1) - 1))^2 + 5((2(3|x - 2| + 1) + 2)/(-2(3|x - 2| + 1) - 1)) - 3) - 1) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (3(-2(-1x - 5)^2 + 5) + 3)/(1(-2(-1x - 5)^2 + 5) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(1sin(2\u03c0(-2x + 1) + 1)) - 2)/(3(1sin(2\u03c0(-2x + 1) + 1)) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2sin(2\u03c0((-3(1|(-4x^2 + 2x + 4) - 3| - 2) + 2)/(-2(1|(-4x^2 + 2x + 4) - 3| - 2) - 3)) + 3) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1|(-2cos(1\u03c0(4((1x)/(1x + 1))^2 + 3((1x)/(1x + 1)) - 3) - 2) - 2)| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1(-1cos(1\u03c0(1|x - 1| - 3) + 3)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(2\u03c0(2(1|x + 1| + 1)^2 + 3(1|x + 1| + 1) + 2) - 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "293", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-1|(-2sin(1\u03c0x + 1))| + 2)^2 - 4(-1|(-2sin(1\u03c0x + 1))| + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(-1(1cos(2\u03c0x - 1) + 3) - 5) + 1)/(2(-1(1cos(2\u03c0x - 1) + 3) - 5) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(2|(1sin(2\u03c0x - 2) + 2) + 3| - 2)^2 - 1(2|(1sin(2\u03c0x - 2) + 2) + 3| - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3sin(2\u03c0(-4((2x)/(3x + 2))^2 - 3((2x)/(3x + 2)) - 5) - 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2|((-1(3(3x^2 - 3x - 4) - 4) - 1)/(-3(3(3x^2 - 3x - 4) - 4) - 3))| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "174", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(2cos(1\u03c0(1|((-2x + 3)/(-2x - 3)) + 3| - 3)) + 3) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1sin(1\u03c0(5(3|x + 3|) + 4) + 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "301", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(-1(-1|(2cos(3\u03c0x + 2)) + 3| + 2)^2 + 2(-1|(2cos(3\u03c0x + 2)) + 3| + 2)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(1\u03c0(-1(1|((-2x)/(2x - 3)) - 3| + 3)^2) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2(3(1cos(3\u03c0x) - 1)^2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4((2(-2|(-2cos(2\u03c0x + 2) + 2) - 3| - 3))/(-2(-2|(-2cos(2\u03c0x + 2) + 2) - 3| - 3) - 3))^2 + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3cos(3\u03c0(3|(-1x) - 2| - 1) - 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3|((1(-1sin(1\u03c0x - 3) - 1) - 3)/(3(-1sin(1\u03c0x - 3) - 1) + 3)) - 2| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3|(5(-2(-3sin(2\u03c0x + 2) - 1) - 1)^2 - 1(-2(-3sin(2\u03c0x + 2) - 1) - 1) - 3) - 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1sin(2\u03c0(5(-2x^2 - 4x - 4) - 2)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "499", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-3(4(-3sin(3\u03c0(1x + 4) + 3))^2 + 2(-3sin(3\u03c0(1x + 4) + 3))) + 3)/(2(4(-3sin(3\u03c0(1x + 4) + 3))^2 + 2(-3sin(3\u03c0(1x + 4) + 3))) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(1|((1(-2sin(1\u03c0x - 2) + 2))/(3(-2sin(1\u03c0x - 2) + 2) - 2))| + 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1cos(3\u03c0((2(-4x^2 - 5x + 4) - 2)/(1(-4x^2 - 5x + 4)))) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3sin(1\u03c0((1(-1x - 1) + 3)/(-3(-1x - 1) + 2)) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1|(-3cos(2\u03c0(-2x + 1) + 2)) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "159", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3sin(3\u03c0((-2(1(2x^2 - 2x) - 5))/(-1(1(2x^2 - 2x) - 5) + 2)) + 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(2sin(1\u03c0(1x^2 + 2x + 4) + 1) + 3) + 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "203", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|((1(1(1cos(1\u03c0x - 3) - 3) - 4) - 2)/(2(1(1cos(1\u03c0x - 3) - 3) - 4))) - 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 5(-2cos(3\u03c0((-3x + 1)/(3x - 2)) + 3))^2 - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1cos(2\u03c0(1|(-5x + 5) - 2| - 1)) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "199", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(1\u03c0(3|((2x - 1)/(3x + 2)) - 3| + 1) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1(-4(-2cos(1\u03c0x - 3) - 1)^2 - 1(-2cos(1\u03c0x - 3) - 1) - 4) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(3cos(3\u03c0(3|x + 2| + 1) - 3) - 1) + 2)/(-2(3cos(3\u03c0(3|x + 2| + 1) - 3) - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4(-3|((2(1sin(1\u03c0x) - 2) - 3)/(-3(1sin(1\u03c0x) - 2) - 2)) + 3| + 3)^2 + 3(-3|((2(1sin(1\u03c0x) - 2) - 3)/(-3(1sin(1\u03c0x) - 2) - 2)) + 3| + 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1|(-1sin(3\u03c0(5x - 1) - 1)) + 1| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "150", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1(3|(2cos(3\u03c0((-3x - 3)/(-3x + 1))) - 3) + 1| + 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(1|(-1(-3cos(1\u03c0x - 1) - 2)^2 + 5(-3cos(1\u03c0x - 1) - 2) - 5) + 1| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 5(-2cos(2\u03c0(1x^2 + 5x - 4) + 2) - 1) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "210", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-2(-3(1sin(3\u03c0(-2x^2 + 1x - 5) - 3) + 3) - 3) - 1)/(2(-3(1sin(3\u03c0(-2x^2 + 1x - 5) - 3) + 3) - 3) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "260", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(-3(2cos(1\u03c0((1x - 2)/(1x))) - 3)^2 + 4(2cos(1\u03c0((1x - 2)/(1x))) - 3) + 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1sin(3\u03c0(1((-2(-3x^2 + 4) + 3)/(1(-3x^2 + 4))) - 5) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3(1|((1(1sin(2\u03c0x) + 2) - 3)/(3(1sin(2\u03c0x) + 2) + 1))| - 3)^2 + 3(1|((1(1sin(2\u03c0x) + 2) - 3)/(3(1sin(2\u03c0x) + 2) + 1))| - 3) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2sin(3\u03c0(-1((-3(5x - 1) + 3)/(1(5x - 1) + 1))^2 - 5((-3(5x - 1) + 3)/(1(5x - 1) + 1)) + 5) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -5(-1sin(1\u03c0(1|((-2x - 1)/(-1x + 2)) - 1| + 2) + 1) + 2)^2 - 3(-1sin(1\u03c0(1|((-2x - 1)/(-1x + 2)) - 1| + 2) + 1) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|(-1cos(3\u03c0(-4(3x^2 - 5x) + 5) - 1) + 3) - 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "485", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2|(-1sin(2\u03c0(3x - 5) + 2) - 2) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "226", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1(1sin(2\u03c0((-2x)/(2x + 2)) - 2) - 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2sin(3\u03c0(-2(-3x^2 + 1x - 1) + 5) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "251", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (2(3(-3(3sin(3\u03c0x - 1) + 3) - 1)^2 - 3) - 2)/(3(3(-3(3sin(3\u03c0x - 1) + 3) - 1)^2 - 3) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2((-1(-1sin(2\u03c0(-3x - 4) - 1) - 3))/(-1(-1sin(2\u03c0(-3x - 4) - 1) - 3) + 2))^2 + 2((-1(-1sin(2\u03c0(-3x - 4) - 1) - 3))/(-1(-1sin(2\u03c0(-3x - 4) - 1) - 3) + 2)) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (3(-1|(-2sin(2\u03c0x + 1) + 1) - 2| + 3) + 2)/(-3(-1|(-2sin(2\u03c0x + 1) + 1) - 2| + 3) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2cos(1\u03c0(-4(-2|x + 1| + 3) - 1)) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "159", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1cos(3\u03c0((-2(-3x^2 - 3x + 3) + 3)/(-3(-3x^2 - 3x + 3) - 2)) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3(2|(-1cos(3\u03c0x - 1) + 2) - 2| + 2) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(1\u03c0(3|(-3x + 2) + 2| - 1) - 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5(1|(3sin(1\u03c0(-4x^2 - 2x - 3) - 1) + 1) + 2| - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "575", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(1|(3x + 4) + 2|) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(-3(-3cos(1\u03c0((-2x - 1)/(1x - 3)))) - 1) - 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-2(-1cos(1\u03c0x - 1) + 2) - 3) + 1)/(1(-2(-1cos(1\u03c0x - 1) + 2) - 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2cos(2\u03c0(-4(5((1x)/(3x - 2))^2 - 2((1x)/(3x - 2)) + 4) + 5) - 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(3(-1(-3e^(-2x) - 2) + 3)^2 + 4(-1(-3e^(-2x) - 2) + 3) - 5) - 3)/(3(3(-1(-3e^(-2x) - 2) + 3)^2 + 4(-1(-3e^(-2x) - 2) + 3) - 5) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1(2sin(1\u03c0(-1x + 1) - 2) + 3)^2 + 4(2sin(1\u03c0(-1x + 1) - 2) + 3) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|(2cos(1\u03c0(5x - 4)) + 3)| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3(5(-1|(1cos(2\u03c0x - 2) + 2) - 3| + 2)^2 + 5(-1|(1cos(2\u03c0x - 2) + 2) - 3| + 2) - 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3(-1(2sin(2\u03c0(3|x - 3| + 1) - 1))^2 + 2(2sin(2\u03c0(3|x - 3| + 1) - 1)) + 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(3(2cos(3\u03c0(-1|x|) - 3) + 2)^2 - 5(2cos(3\u03c0(-1|x|) - 3) + 2) - 5) + 3)/(-3(3(2cos(3\u03c0(-1|x|) - 3) + 2)^2 - 5(2cos(3\u03c0(-1|x|) - 3) + 2) - 5) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(3sin(1\u03c0(-5x^2 + 3x) + 3) - 3) + 2)/(-1(3sin(1\u03c0(-5x^2 + 3x) + 3) - 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "497", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|(1cos(1\u03c0(-1x - 4)) - 1) - 1| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(3\u03c0(3|((-3x - 2)/(-1x + 3)) + 1| + 3) - 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-5(1cos(2\u03c0(-5x)))^2 + 5(1cos(2\u03c0(-5x))) - 2))/(3(-5(1cos(2\u03c0(-5x)))^2 + 5(1cos(2\u03c0(-5x))) - 2) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "400", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (1(2sin(3\u03c0(5(1|x + 3| - 3)^2 - 3(1|x + 3| - 3) + 2)) + 2))/(-3(2sin(3\u03c0(5(1|x + 3| - 3)^2 - 3(1|x + 3| - 3) + 2)) + 2) - 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "382", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2cos(1\u03c0(-3|(4((-2x + 3)/(1x + 1)) - 2) - 3| + 2) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(2|(-2sin(2\u03c0x + 2) + 1) - 3| + 3)^2 + 4(2|(-2sin(2\u03c0x + 2) + 1) - 3| + 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4(1((1(1cos(1\u03c0x + 3) + 2) - 3)/(-2(1cos(1\u03c0x + 3) + 2) - 3))^2 + 5((1(1cos(1\u03c0x + 3) + 2) - 3)/(-2(1cos(1\u03c0x + 3) + 2) - 3)) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3cos(1\u03c0(3|(-1((-2x + 3)/(-3x - 1)) + 3) - 1| - 1) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(-2(2sin(3\u03c0x - 3) - 2) + 3)^2 - 4(-2(2sin(3\u03c0x - 3) - 2) + 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(3cos(1\u03c0(-3x - 1) + 1) + 1) + 1)/(3(3cos(1\u03c0(-3x - 1) + 1) + 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(3\u03c0(-2|((1(2x^2 + 5x - 4) - 1)/(1(2x^2 + 5x - 4))) + 1| + 3) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 4(-1sin(1\u03c0(3|((-2x + 1)/(3x + 3)) - 3| + 2)) - 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(2(-1cos(2\u03c0(-2x^2 - 2x - 4) - 2) + 3) - 5) + 2| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "286", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(3\u03c0(-2(-1x^2 + 3x + 3)) + 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2(-2|(2sin(2\u03c0((2x - 1)/(-1x - 1)) + 1) - 1) - 3| - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2|((2(-1cos(1\u03c0x) - 2) - 3)/(1(-1cos(1\u03c0x) - 2) - 2)) + 1| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1|(2(-3sin(1\u03c0x - 3) - 3) + 1)| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1|(-4(1sin(3\u03c0(3x + 4) + 2) - 3)^2 + 1(1sin(3\u03c0(3x + 4) + 2) - 3) + 2) - 3| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3sin(2\u03c0(4(1|(-1x - 4)|)^2 - 2(1|(-1x - 4)|) + 5) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "483", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|((1(3cos(3\u03c0(-5x) + 1) + 1) + 3)/(-1(3cos(3\u03c0(-5x) + 1) + 1) - 2)) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-3|(1cos(2\u03c0x + 1) - 2) - 2| - 1) - 1)/(-3(-3|(1cos(2\u03c0x + 1) - 2) - 2| - 1) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "185", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1sin(3\u03c0(3|(-4x - 4) - 3| + 2) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "719", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1(1|(3cos(3\u03c0((-2x - 3)/(1x)) + 1) + 3) + 3| - 1)^2 + 3(1|(3cos(3\u03c0((-2x - 3)/(1x)) + 1) + 3) + 3| - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3sin(1\u03c0(3|(-1x + 1)| - 2) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3sin(2\u03c0(1|(-1x + 4) - 3| - 2) - 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(1\u03c0(-1|(4((3x + 2)/(-1x - 3))^2 - 1((3x + 2)/(-1x - 3)) + 5) + 2| - 2) + 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "187", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1sin(2\u03c0(-1((-1(-1x + 3) - 2)/(1(-1x + 3) + 3))^2 + 1((-1(-1x + 3) - 2)/(1(-1x + 3) + 3)) + 1) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3|((-3(3sin(1\u03c0(-4x^2 + 3x - 2) - 1) + 1) - 3)/(2(3sin(1\u03c0(-4x^2 + 3x - 2) - 1) + 1) + 2))| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(2|(-1sin(2\u03c0x + 3) - 2) - 3| + 1)^2 - 2(2|(-1sin(2\u03c0x + 3) - 2) - 3| + 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3|(-1(2(1sin(2\u03c0x + 1) + 2) - 2)^2 + 2) - 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2sin(1\u03c0(3(-2(3|x + 3| + 2)^2 - 2(3|x + 3| + 2) + 3) - 4) + 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "458", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(-2sin(2\u03c0(1(-3|x + 1| - 1) - 5) + 1) - 2) - 1)/(1(-2sin(2\u03c0(1(-3|x + 1| - 1) - 5) + 1) - 2) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 4(-3sin(3\u03c0(-3|x + 2| + 2) + 2) - 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-3(-1cos(1\u03c0(1|x - 2| - 2) - 3) - 2))/(-2(-1cos(1\u03c0(1|x - 2| - 2) - 3) - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3|(-3cos(3\u03c0(3x^2 - 2x + 3) + 3) + 1) + 2| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "260", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(2cos(3\u03c0(-5(-3|x - 2| + 3) + 1) + 3) + 2)^2 - 4(2cos(3\u03c0(-5(-3|x - 2| + 3) + 1) + 3) + 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "283", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4(-3(-3cos(1\u03c0x - 1) + 3) + 2)^2 + 2(-3(-3cos(1\u03c0x - 1) + 3) + 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2\u03c0(4(5x + 1)^2 - 1(5x + 1) - 1)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "408", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2|(-3sin(1\u03c0(-3x + 1) + 1) + 3)| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1|(2((-1(-1sin(1\u03c0x + 2) + 2) - 2)/(2(-1sin(1\u03c0x + 2) + 2) - 1)) - 5) - 3| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(1\u03c0(-4(-1|x - 2|)^2 + 5(-1|x - 2|) + 3) + 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "275", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0((-3(-1|x|) - 1)/(-2(-1|x|) - 1)) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-3(2cos(3\u03c0(-2x - 3) + 1) + 2) + 1)/(3(2cos(3\u03c0(-2x - 3) + 1) + 2)) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(-3sin(3\u03c0(2(-2x - 5)^2 - 4(-2x - 5) + 4) + 3) + 3) - 2)/(2(-3sin(3\u03c0(2(-2x - 5)^2 - 4(-2x - 5) + 4) + 3) + 3) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(2sin(1\u03c0((-1(-5x - 3) + 3)/(-1(-5x - 3)))) + 3) - 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(3\u03c0(4(5x^2 - 1x + 2) + 5) + 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(1sin(3\u03c0(-3((-3x + 2)/(-2x - 2)) + 4)) + 3) + 3| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1|(-3((3(-3sin(2\u03c0x - 2) + 1) + 2)/(-2(-3sin(2\u03c0x - 2) + 1) - 2))^2 - 5((3(-3sin(2\u03c0x - 2) + 1) + 2)/(-2(-3sin(2\u03c0x - 2) + 1) - 2)) - 3) - 3| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2cos(3\u03c0((3(2x - 5) + 2)/(-1(2x - 5) + 3)) - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 4(2sin(3\u03c0(-2x - 5) - 1) - 1)^2 - 4(2sin(3\u03c0(-2x - 5) - 1) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3sin(2\u03c0(2|(-4(1x - 1)^2 - 3) - 1| - 2) + 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "503", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3|(1cos(3\u03c0(-1x + 5) + 3) + 3) + 3| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|((-1(2(-3cos(3\u03c0x) - 1) + 3) - 3)/(3(2(-3cos(3\u03c0x) - 1) + 3) - 2)) + 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3(-3|(-1cos(3\u03c0x + 3) - 3) - 1| - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1(5(-2cos(1\u03c0(-2|x - 1| - 3) + 3)) + 5)^2 - 2(5(-2cos(1\u03c0(-2|x - 1| - 3) + 3)) + 5) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2cos(1\u03c0((-3(3|x + 2| - 1) + 1)/(1(3|x + 2| - 1) + 2)) + 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2((-2(-1cos(2\u03c0x - 2) + 3) - 2)/(-2(-1cos(2\u03c0x - 2) + 3) + 1))^2 + 1((-2(-1cos(2\u03c0x - 2) + 3) - 2)/(-2(-1cos(2\u03c0x - 2) + 3) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 4(-3sin(3\u03c0((3x - 2)/(1x - 3)) - 2) - 2)^2 - 2(-3sin(3\u03c0((3x - 2)/(1x - 3)) - 2) - 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-1(3cos(1\u03c0(-1x + 2)) - 3))/(-3(3cos(1\u03c0(-1x + 2)) - 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(-2e^(-2(2|x - 2| - 1)) - 2) + 2)/(1(-2e^(-2(2|x - 2| - 1)) - 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(2\u03c0(2(3((1x - 2)/(3x - 1))^2 - 2((1x - 2)/(3x - 1)) - 5) + 3) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3(-1sin(3\u03c0(-2|x - 1| - 3) + 3))^2 - 2(-1sin(3\u03c0(-2|x - 1| - 3) + 3)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 5(-2sin(3\u03c0(1x - 2)) - 1)^2 - 4(-2sin(3\u03c0(1x - 2)) - 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1(1cos(2\u03c0((1x - 1)/(-1x - 3)) + 1) - 3)^2 - 1(1cos(2\u03c0((1x - 1)/(-1x - 3)) + 1) - 3) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1cos(1\u03c0(5((-1(-5x^2 - 3x + 3) - 1)/(-3(-5x^2 - 3x + 3))) + 5) - 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3|(-1cos(1\u03c0(-3(1x^2 - 3x - 5) - 5) + 2)) + 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "555", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2(-1sin(2\u03c0(-2x + 2) - 3))^2 - 4(-1sin(2\u03c0(-2x + 2) - 3)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1|(1(2sin(3\u03c0x + 3)) - 4) - 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1|(2e^(2((-3x - 3)/(-1x - 1))) - 3)| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1|((-2(-4(3sin(1\u03c0x - 2) + 1)^2 + 4(3sin(1\u03c0x - 2) + 1) + 5) + 1)/(-1(-4(3sin(1\u03c0x - 2) + 1)^2 + 4(3sin(1\u03c0x - 2) + 1) + 5) - 1)) - 2| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1sin(3\u03c0(-2|((-1x + 1)/(-1x)) - 2| - 2) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1|(3(2cos(3\u03c0x - 3) - 2) + 4) - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3cos(1\u03c0(-2|(-3(5x - 4)^2 + 1(5x - 4) - 4)| - 3) + 3) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "222", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (2(3|(-1e^(2(4x + 1))) + 1| - 3) - 2)/(3(3|(-1e^(2(4x + 1))) + 1| - 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(-1|(-1sin(1\u03c0x) - 3)|) - 2)/(3(-1|(-1sin(1\u03c0x) - 3)|) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5((-3(1e^(-2x) - 2) - 3)/(2(1e^(-2x) - 2) + 2)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "304", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 5(1cos(1\u03c0(4((-2x + 1)/(3x + 1))^2 + 3) - 1) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4((-2(2cos(1\u03c0(-3x - 3) + 2) - 1) + 2)/(-1(2cos(1\u03c0(-3x - 3) + 2) - 1) + 3))^2 - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2sin(2\u03c0(-3|(4x + 5) + 2| - 1) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(1\u03c0(-3(-1|((-2x - 1)/(-1x - 1)) - 2| - 3)^2 + 2(-1|((-2x - 1)/(-1x - 1)) - 2| - 3) + 1) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(1cos(2\u03c0(-1|x + 2| + 2) + 2) - 1) - 3)/(-1(1cos(2\u03c0(-1|x + 2| + 2) + 2) - 1) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2\u03c0(-1|(-1(3x^2 - 5x + 3) - 1) + 3|) + 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "461", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(1\u03c0(-1|(2(1x^2 - 3x - 2) + 5)| - 2) + 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "411", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(2\u03c0(-3(-1|((-1x - 1)/(3x + 2)) + 2| + 3)^2 + 1) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(2\u03c0(-2|(5x - 2) + 3| + 2) - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "401", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(-2cos(1\u03c0(-3(3x + 4)^2 - 2(3x + 4) + 3) + 1) + 2) - 1| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "464", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1cos(3\u03c0(-1(-1|x + 2| - 1) - 4) + 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(4(1sin(3\u03c0x + 1) + 3)^2 + 5) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (1(2(2x^2 + 3x - 2) - 4) + 1)/(3(2(2x^2 + 3x - 2) - 4) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1(-1|(3sin(1\u03c0(-1x^2 - 2x + 2) + 3) - 1) + 1| + 2) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1sin(1\u03c0(-3(2|(2x^2 - 4x + 4)| - 2) - 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "250", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3sin(2\u03c0(1((2x + 2)/(3x + 1)) + 5) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2sin(2\u03c0(-2|(4((2x + 3)/(2x - 1)) + 3) - 1|) + 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(3\u03c0(-2|((-1(-3x^2 + 3) - 3)/(-2(-3x^2 + 3))) - 3| + 2) - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 1|(-3(-1sin(3\u03c0x - 1) + 2)) - 1| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1cos(2\u03c0(2(-1|x - 1| - 3)^2 - 4(-1|x - 1| - 3) - 2) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "260", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-1cos(3\u03c0(2((2x - 1)/(-1x + 2))^2 + 5((2x - 1)/(-1x + 2)) + 3) + 3) - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (3(-1e^(-1(-3x - 2)) - 3) - 3)/(2(-1e^(-1(-3x - 2)) - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2sin(2\u03c0(-5(-4x^2 + 4x + 4) - 4) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "482", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3cos(2\u03c0(5(-1((-2x - 2)/(1x)) + 5)^2 - 4(-1((-2x - 2)/(1x)) + 5) - 4) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "283", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5(2sin(1\u03c0(2|x - 3| - 1)) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3|(-2sin(1\u03c0(-2x^2 + 5x - 5) + 2) - 1) - 3| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "204", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(2\u03c0((3(2x - 1) + 3)/(-1(2x - 1))) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(1(-2sin(3\u03c0(2x^2 - 2x - 4) + 2) - 1)) - 3| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (2(-2cos(2\u03c0(3|(5x - 5) - 1|) + 3) + 3) + 2)/(2(-2cos(2\u03c0(3|(5x - 5) - 1|) + 3) + 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "599", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-1(-2(-2cos(1\u03c0x) + 3) + 1) - 1)/(-1(-2(-2cos(1\u03c0x) + 3) + 1) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3sin(1\u03c0(1|(-1x + 3)|) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1|(5(-2cos(3\u03c0(-4x^2 - 1x + 5) - 1) - 3) + 4) - 3| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(2cos(3\u03c0(-1x^2 + 5) - 3) - 3) - 1| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "569", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (3(-3e^(-2(-2|(1x - 1) - 2| + 2)) - 3) - 3)/(3(-3e^(-2(-2|(1x - 1) - 2| + 2)) - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 5((-2(1(3sin(1\u03c0x) - 2)^2 + 1(3sin(1\u03c0x) - 2) - 4) - 1)/(1(1(3sin(1\u03c0x) - 2)^2 + 1(3sin(1\u03c0x) - 2) - 4) + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -4(-3cos(3\u03c0((-1(5x^2 + 4x - 2) - 2)/(-3(5x^2 + 4x - 2) - 3)) + 1) - 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2sin(2\u03c0((1(-1|x + 1| + 2) - 3)/(-2(-1|x + 1| + 2) + 3))) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(2\u03c0(-2(-2|x - 1| - 2)^2 + 3) - 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "263", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1((2(2cos(3\u03c0x - 2) + 2) - 2)/(-2(2cos(3\u03c0x - 2) + 2) - 3))^2 + 2((2(2cos(3\u03c0x - 2) + 2) - 2)/(-2(2cos(3\u03c0x - 2) + 2) - 3)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -4(-3(-2sin(3\u03c0((-2x + 1)/(-3x + 4)) - 1) + 3)^2 + 4(-2sin(3\u03c0((-2x + 1)/(-3x + 4)) - 1) + 3) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1(-1sin(1\u03c0((-1x + 3)/(-1x + 2)) + 1)) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(-4((3(-2cos(3\u03c0x + 2) + 2) - 1)/(1(-2cos(3\u03c0x + 2) + 2) + 1)) - 2) + 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-1(2|(-1cos(2\u03c0x + 2) + 3) + 1| - 1) - 2)/(2(2|(-1cos(2\u03c0x + 2) + 3) + 1| - 1) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(3\u03c0(-3|(2x - 2) + 3| + 3) - 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "179", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3((1(-3cos(2\u03c0x + 3) + 3) - 1)/(1(-3cos(2\u03c0x + 3) + 3))) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1((-2(-1sin(3\u03c0x - 2) + 1) - 3)/(1(-1sin(3\u03c0x - 2) + 1) - 3)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 4(3sin(1\u03c0(2(-3|x + 3|)^2 + 4) + 1)) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "246", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|(2sin(1\u03c0(5x^2 + 1x + 2) - 2) - 1) + 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "509", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1cos(3\u03c0(5(-3|x + 2| - 2)^2 - 2(-3|x + 2| - 2) + 5) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "439", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3cos(2\u03c0((1(5x + 2) - 1)/(1(5x + 2) + 2)) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(3\u03c0(-4((3(3|x|) - 3)/(-2(3|x|) - 2)) - 1) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(-1(2sin(2\u03c0(-2x^2 - 2x + 1) + 1) + 1) + 3) + 1)/(3(-1(2sin(2\u03c0(-2x^2 - 2x + 1) + 1) + 1) + 3) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "569", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(-2sin(3\u03c0(3|x - 1| + 3)) - 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-1(1sin(3\u03c0(1x^2 - 3x - 3) + 2) + 3)) - 1)/(-1(-1(1sin(3\u03c0(1x^2 - 3x - 3) + 2) + 3)) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "552", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-2(-1cos(2\u03c0(-3(-5x^2 + 4x - 1) + 3) - 2) - 3) + 3)/(-1(-1cos(2\u03c0(-3(-5x^2 + 4x - 1) + 3) - 2) - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "290", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(1(-2sin(1\u03c0(-3x + 3)) - 2)^2 - 2(-2sin(1\u03c0(-3x + 3)) - 2) + 3) + 2)/(2(1(-2sin(1\u03c0(-3x + 3)) - 2)^2 - 2(-2sin(1\u03c0(-3x + 3)) - 2) + 3) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(2sin(3\u03c0((-2x + 1)/(1x + 2)) + 2) - 2) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3(-2cos(2\u03c0((-2x)/(-3x + 4)) - 1) + 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -5(-5(-3cos(3\u03c0x - 3) - 1) - 3)^2 - 3(-5(-3cos(3\u03c0x - 3) - 1) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -4((-2(1|(3sin(2\u03c0x - 2) + 1) - 1| - 2) + 3)/(-2(1|(3sin(2\u03c0x - 2) + 1) - 1| - 2) - 1)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5(-2((2(2sin(1\u03c0x - 2) + 3))/(-1(2sin(1\u03c0x - 2) + 3) - 1))^2 - 1((2(2sin(1\u03c0x - 2) + 3))/(-1(2sin(1\u03c0x - 2) + 3) - 1))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2(3(2|(2sin(1\u03c0x - 3) + 2) - 3| - 1) + 2)^2 + 5(3(2|(2sin(1\u03c0x - 3) + 2) - 3| - 1) + 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5(3|(2cos(1\u03c0x + 2) + 3) + 3| + 1) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1(3(2cos(3\u03c0x) - 2)^2 + 2(2cos(3\u03c0x) - 2) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (1(-2(2|(1cos(3\u03c0x - 1) + 2)| - 1) - 5) - 1)/(2(-2(2|(1cos(3\u03c0x - 1) + 2)| - 1) - 5) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2(-1|(3sin(3\u03c0x - 2) + 1) - 1| - 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "110", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3cos(2\u03c0(2(-1|x - 3|) + 1) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-5(1|(-1cos(1\u03c0x - 3)) - 2| + 3)^2 - 5) - 1)/(1(-5(1|(-1cos(1\u03c0x - 3)) - 2| + 3)^2 - 5)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (2(3sin(1\u03c0(5(3x^2 - 3x - 1) - 3) - 1) + 2) + 2)/(1(3sin(1\u03c0(5(3x^2 - 3x - 1) - 3) - 1) + 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "197", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(-1sin(3\u03c0(-1x^2 + 4x - 1) - 1) + 3) + 2)/(-3(-1sin(3\u03c0(-1x^2 + 4x - 1) - 1) + 3) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "332", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3(3cos(1\u03c0(-1x - 5) + 1))^2 - 3(3cos(1\u03c0(-1x - 5) + 1)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (2(-1|(3sin(1\u03c0(-4x + 1) - 2) - 2) - 3| + 1) - 3)/(2(-1|(3sin(1\u03c0(-4x + 1) - 2) - 2) - 3| + 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3(-2cos(1\u03c0((-1x - 1)/(2x - 1)) - 3) - 1)^2 - 3(-2cos(1\u03c0((-1x - 1)/(2x - 1)) - 3) - 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2((-2(-1sin(3\u03c0x + 1) + 2) + 3)/(3(-1sin(3\u03c0x + 1) + 2) - 2))^2 + 2((-2(-1sin(3\u03c0x + 1) + 2) + 3)/(3(-1sin(3\u03c0x + 1) + 2) - 2)) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(4((-2x - 3)/(-1x))) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1|(1cos(1\u03c0(1x + 4) - 3) - 1) - 1| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}