{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|((-3(-2sin(1\u03c0x - 2) + 2) + 2)/(2(-2sin(1\u03c0x - 2) + 2) + 1)) - 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) = -2sin(2\u03c0(2|(3x^2 - 2x + 5) - 1|) + 1) - 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) = -3|(-4(-3x^2 - 2x)^2 - 4(-3x^2 - 2x) + 1) + 1| + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1sin(3\u03c0(-4(1x - 1) - 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) = -2sin(3\u03c0(-4(-4x + 2)^2 + 3(-4x + 2) - 3)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "390", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2sin(1\u03c0(3(-5x + 4)^2 - 2(-5x + 4) + 4) - 3) + 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": "Determine the count of local maxima for the function f(x) = -1sin(3\u03c0(-1(-4x^2 + 3x - 2) + 5) - 2) + 2 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) = -2(2e^(-1(-5x + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(1\u03c0(-2|(-1|x - 3| - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(1\u03c0(1(5x - 2) - 4) + 1) + 1 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 minimum points for f(x) = -1|(-1|(-1cos(1\u03c0x + 2) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2|(2cos(2\u03c0(2x + 3) + 1) + 3) - 1| - 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 minimum points for f(x) = -3(1sin(3\u03c0(3x^2 - 5x) - 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "259", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(3\u03c0(3(2|x + 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": "361", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -3cos(3\u03c0(1(-2x^2 - 2x + 2)^2 - 2(-2x^2 - 2x + 2) - 2) - 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": "Find the number of local maximum points for f(x) = -1|((-3(2cos(3\u03c0x - 1) - 3) - 1)/(-2(2cos(3\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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2|(-1cos(3\u03c0(5x^2 + 4) + 3) - 3) + 1| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "248", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1(2cos(3\u03c0(2x + 1) - 2) + 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": "Determine the count of local maxima for the function f(x) = 2(1sin(3\u03c0(2x + 5) + 1) - 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": "Determine the count of local minima for the function f(x) = -3(-1cos(2\u03c0((2x - 1)/(-2x - 1)) + 1) - 3)^2 - 3(-1cos(2\u03c0((2x - 1)/(-2x - 1)) + 1) - 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": "Determine the count of local maxima for the function f(x) = 3|(5(-3cos(2\u03c0x - 1) + 2)^2 + 1(-3cos(2\u03c0x - 1) + 2) - 3)| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "159", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(-2cos(2\u03c0(-4x^2 + 5x + 1) - 2) - 3)^2 + 1(-2cos(2\u03c0(-4x^2 + 5x + 1) - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "517", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3|((1(3cos(2\u03c0x - 2) + 1) + 2)/(2(3cos(2\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": "39", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(2cos(2\u03c0((3x)/(1x + 1)) - 1) - 3)^2 - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-3((3(2sin(1\u03c0x + 2) + 3))/(-3(2sin(1\u03c0x + 2) + 3) + 4)) + 1)/(-1((3(2sin(1\u03c0x + 2) + 3))/(-3(2sin(1\u03c0x + 2) + 3) + 4))) 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) = 3sin(3\u03c0(1((-3x - 2)/(-3x + 1))^2 + 4((-3x - 2)/(-3x + 1)) + 4) - 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) = -3|(-2cos(3\u03c0(-3x - 2) - 3) - 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": "Determine the count of local maxima for the function f(x) = -3cos(1\u03c0(-3((2x - 2)/(-1x + 3)) - 4) + 3) - 1 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 extrema (minima and maxima combined) does the function f(x) = (1(-2(-3cos(1\u03c0x + 2) + 1)^2 - 3(-3cos(1\u03c0x + 2) + 1) - 1) - 3)/(1(-2(-3cos(1\u03c0x + 2) + 1)^2 - 3(-3cos(1\u03c0x + 2) + 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": "Find the number of local maximum points for f(x) = 4(-5(-1sin(3\u03c0x - 1) + 3)^2 - 1(-1sin(3\u03c0x - 1) + 3) - 1) + 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 total count of local extrema points for the function f(x) = (-1(1|(-1cos(2\u03c0x + 1) + 3)|) + 3)/(-2(1|(-1cos(2\u03c0x + 1) + 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) = -1cos(2\u03c0((3(-3|x + 3| - 1) - 3)/(-1(-3|x + 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": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1sin(1\u03c0(4(4x + 1)^2 + 5(4x + 1) - 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|(2(1|x + 2|)^2 - 4(1|x + 2|) - 2) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1|(-3cos(2\u03c0(4x^2 - 3) - 1) + 2) + 2| - 2 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": "Find the total number of local minima and maxima for f(x) = -3(1(2cos(3\u03c0x + 2) + 1) + 4)^2 + 1(1(2cos(3\u03c0x + 2) + 1) + 4) + 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 maxima does the function f(x) = 3(-2sin(1\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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3(1|(-4x^2 - 4x) + 2| - 3)^2 - 1(1|(-4x^2 - 4x) + 2| - 3) + 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": "Find the number of local maximum points for f(x) = 1(2cos(1\u03c0((1x + 3)/(2x)) - 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": "How many local maxima does the function f(x) = 1sin(3\u03c0(-3((-2x + 2)/(2x - 1))^2 + 3((-2x + 2)/(2x - 1)) - 5) + 1) - 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 minimum points for f(x) = 3(-5(-3sin(1\u03c0x + 2) + 2) - 4) - 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 count of local minima for the function f(x) = (-1(-2cos(1\u03c0(-3|x + 2| - 1) + 3) - 1) - 1)/(-3(-2cos(1\u03c0(-3|x + 2| - 1) + 3) - 1) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "117", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(-1|(3cos(2\u03c0x + 1) - 3) - 1| - 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) = (-3(1cos(1\u03c0(-1|x - 1|) - 3)) + 3)/(3(1cos(1\u03c0(-1|x - 1|) - 3)) - 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 number of local minimum points for f(x) = -2sin(2\u03c0(3(2x + 3) + 4) - 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 count of local minima for the function f(x) = -1((-3(1sin(1\u03c0x + 3)) - 2)/(3(1sin(1\u03c0x + 3))))^2 + 1((-3(1sin(1\u03c0x + 3)) - 2)/(3(1sin(1\u03c0x + 3)))) - 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 extrema (minima and maxima combined) does the function f(x) = -3cos(2\u03c0(3(3x + 4) + 1) - 2) - 1 have 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 maxima does the function f(x) = 5(1|(-3cos(1\u03c0x + 3)) - 1| - 3) + 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": "Find the number of local maximum points for f(x) = 5(1cos(2\u03c0(1|x + 1| - 2) - 2) + 2) - 4 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) = -1|(2e^(1(1x^2 + 2x - 5)) - 1) - 2| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1(-3(-4x^2 + 1x + 4)^2 + 3(-4x^2 + 1x + 4) - 1)^2 - 4(-3(-4x^2 + 1x + 4)^2 + 3(-4x^2 + 1x + 4) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -5(-2cos(1\u03c0(-3|x + 2| + 3) + 3))^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": "Find the total number of local minima and maxima for f(x) = (1(-1sin(1\u03c0(2|x - 2| + 1) + 3) + 3) + 2)/(1(-1sin(1\u03c0(2|x - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(-1sin(3\u03c0(-1x^2 - 3x - 3) - 2) - 1) + 1| + 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) = -3cos(1\u03c0(-2(-4x^2 - 1x + 2) - 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "263", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(2(-2sin(1\u03c0x) + 1) + 2)^2 + 2(2(-2sin(1\u03c0x) + 1) + 2) - 5 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) = (-2(3sin(2\u03c0(-1x) - 1) - 3) + 1)/(1(3sin(2\u03c0(-1x) - 1) - 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": "How many local maxima does the function f(x) = -2sin(3\u03c0(3(-1|x - 3| - 1)^2 - 3) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "262", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(2\u03c0(-1|(2x + 3) - 1| + 1) + 1) - 1 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 count of local maxima for the function f(x) = 2(2|(3cos(3\u03c0x + 3) + 2) + 3| + 1)^2 - 1(2|(3cos(3\u03c0x + 3) + 2) + 3| + 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": "Determine the count of local maxima for the function f(x) = (-1(1cos(3\u03c0(2x) + 3) + 2) - 1)/(-3(1cos(3\u03c0(2x) + 3) + 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": "Determine the count of local minima for the function f(x) = -5(1sin(1\u03c0(-2x^2 - 5x + 5) + 2) - 2)^2 - 4(1sin(1\u03c0(-2x^2 - 5x + 5) + 2) - 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "192", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 4(3|(3sin(2\u03c0x + 1) + 2) + 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": "Determine the count of local maxima for the function f(x) = (1(1sin(3\u03c0(3x + 2) - 1) - 3) - 3)/(-3(1sin(3\u03c0(3x + 2) - 1) - 3) + 4) 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) = 2cos(1\u03c0(5(-2|x - 1|) - 1)) - 1 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 minima does the function f(x) = -5(1sin(1\u03c0(1x^2 + 2x - 2) + 3) + 3)^2 - 3(1sin(1\u03c0(1x^2 + 2x - 2) + 3) + 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2|(-4(2sin(3\u03c0x + 1) - 3)^2 - 2(2sin(3\u03c0x + 1) - 3) + 4) - 2| + 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": "Determine the count of local maxima for the function f(x) = 4(1sin(2\u03c0(-5x - 2) - 1) - 3) + 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": "Find the number of local minimum points for f(x) = -1|(-1(3cos(2\u03c0x - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(-1cos(1\u03c0(1x) - 3) - 3))/(-1(-1cos(1\u03c0(1x) - 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) = -3sin(3\u03c0(-4(2|x + 3| - 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": "238", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(-2|(3cos(1\u03c0x + 2) + 1) - 1| + 1)^2 + 5(-2|(3cos(1\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-2(-3e^(1(2|x - 3| + 1)) + 2))/(3(-3e^(1(2|x - 3| + 1)) + 2) - 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": "Determine the total count of local extrema points for the function f(x) = 5(-4(-1|x + 3| + 3)^2 - 5(-1|x + 3| + 3))^2 - 4(-4(-1|x + 3| + 3)^2 - 5(-1|x + 3| + 3)) + 5 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) = 1(1cos(3\u03c0((-2x - 1)/(2x + 3))) + 1)^2 + 4(1cos(3\u03c0((-2x - 1)/(2x + 3))) + 1) - 4 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) = (-2(-1cos(3\u03c0(3x + 1))))/(2(-1cos(3\u03c0(3x + 1))) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "108", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (3((3(2x - 4) + 3)/(-2(2x - 4) - 2)) + 2)/(1((3(2x - 4) + 3)/(-2(2x - 4) - 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) = -3(3(-1cos(3\u03c0x - 1) + 3)^2 + 5(-1cos(3\u03c0x - 1) + 3) - 4)^2 + 1(3(-1cos(3\u03c0x - 1) + 3)^2 + 5(-1cos(3\u03c0x - 1) + 3) - 4) - 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 total number of local minima and maxima for f(x) = 3|(5(-4x^2 + 2x + 5)^2 - 4) - 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 total count of local extrema points for the function f(x) = -3sin(2\u03c0(2(-5x^2 - 4x - 3) + 2) - 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "493", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -4(1(2sin(3\u03c0x + 3) - 2)^2 + 5(2sin(3\u03c0x + 3) - 2) - 4)^2 - 5(1(2sin(3\u03c0x + 3) - 2)^2 + 5(2sin(3\u03c0x + 3) - 2) - 4) - 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 extrema (minima and maxima combined) does the function f(x) = -1|(-1sin(2\u03c0(-4x - 4) - 1) - 2) - 2| + 2 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) = (-3(-2(2e^(-2x) + 3)^2 + 2(2e^(-2x) + 3) + 1) - 2)/(-2(-2(2e^(-2x) + 3)^2 + 2(2e^(-2x) + 3) + 1) + 1) 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 minimum points for f(x) = (2(3sin(2\u03c0(-5x - 5) + 1) - 1) + 2)/(1(3sin(2\u03c0(-5x - 5) + 1) - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2(-4(2cos(1\u03c0x - 3) - 2)^2 + 1(2cos(1\u03c0x - 3) - 2) - 4) + 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": "Find the total number of local minima and maxima for f(x) = 3cos(2\u03c0(-5(2|x + 3| + 3)) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "401", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(-3(-3cos(2\u03c0x + 3) + 1)) - 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": "Determine the total count of local extrema points for the function f(x) = 1(-2sin(1\u03c0(-5x - 5)) - 1)^2 - 3(-2sin(1\u03c0(-5x - 5)) - 1) - 3 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 total number of local minima and maxima for f(x) = -1sin(1\u03c0((1(1x - 4) + 3)/(3(1x - 4) - 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": "Determine the count of local maxima for the function f(x) = 3|(-1|(2cos(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": "119", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3sin(2\u03c0((-1(-3|x + 2| + 1) + 2)/(3(-3|x + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2|(3cos(2\u03c0(-2x^2 - 1x - 1) + 3)) - 3| + 2 where -10 \u2264 x \u2264 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) = -3(-2sin(3\u03c0(-4x^2 - 2x) - 1) - 1)^2 - 5(-2sin(3\u03c0(-4x^2 - 2x) - 1) - 1) + 1 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 maxima for the function f(x) = 3|(-3sin(1\u03c0(-3x^2 - 4x - 5) - 2) - 2)| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "498", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2cos(2\u03c0((2(-2|x + 2| + 1))/(-3(-2|x + 2| + 1))) - 2) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3(-1|(2cos(2\u03c0x + 1) + 1)| - 1)^2 - 5(-1|(2cos(2\u03c0x + 1) + 1)| - 1) 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": "Determine the count of local minima for the function f(x) = -3cos(1\u03c0(-2(3x^2 + 4) - 3) + 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "225", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(4(-1sin(1\u03c0x - 2) - 2)) + 2)/(2(4(-1sin(1\u03c0x - 2) - 2)) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = (-2(-3sin(2\u03c0(-4x^2 + 3x) + 2) + 2) - 1)/(-1(-3sin(2\u03c0(-4x^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": "523", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3|(2|(-2cos(1\u03c0x + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2|(2cos(3\u03c0(-5x - 5)) - 2)| - 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 minima does the function f(x) = 1(3cos(1\u03c0(-2|x| - 3) - 1) - 3)^2 - 4(3cos(1\u03c0(-2|x| - 3) - 1) - 3) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 2cos(2\u03c0((2(1x + 3) + 2)/(1(1x + 3))) - 1) - 2 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 maxima does the function f(x) = 2cos(2\u03c0(-3|((-2x + 3)/(-2x - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 2sin(2\u03c0(3|(2|x| + 3) + 1| - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3(3(-2|x - 3| + 3)^2 + 4(-2|x - 3| + 3) - 5)^2 - 2(3(-2|x - 3| + 3)^2 + 4(-2|x - 3| + 3) - 5) - 2 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) = -2cos(3\u03c0(3|((2x - 2)/(-1x - 1))| + 1) - 3) - 2 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 total count of local extrema points for the function f(x) = -3((3(1cos(1\u03c0x + 3) + 3) + 2)/(-3(1cos(1\u03c0x + 3) + 3))) + 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 maxima does the function f(x) = 1|(-2(-2|x + 2| + 1)^2 - 5(-2|x + 2| + 1) + 4) - 2| - 2 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 maximum points for f(x) = -3cos(1\u03c0((-2((2x + 3)/(1x + 3)) + 2)/(-1((2x + 3)/(1x + 3)) + 2)) - 1) - 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) = -2cos(1\u03c0((-3((3x + 2)/(1x)) + 3)/(3((3x + 2)/(1x)) - 2)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3(-2cos(2\u03c0((3x - 3)/(1x + 2)) + 2) - 3))/(-1(-2cos(2\u03c0((3x - 3)/(1x + 2)) + 2) - 3) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2cos(1\u03c0(1(4x^2 + 1x + 5)^2 + 1(4x^2 + 1x + 5) + 1) - 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) = 2|(1sin(1\u03c0(-2x^2 - 5x + 4) - 3) - 1) - 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "407", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (2(-3cos(3\u03c0(1x^2 - 1x - 3)) - 2))/(-1(-3cos(3\u03c0(1x^2 - 1x - 3)) - 2) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "563", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 5(-4(-2sin(1\u03c0x + 3) + 1)^2 - 4(-2sin(1\u03c0x + 3) + 1) - 1) - 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 maxima does the function f(x) = -1sin(2\u03c0(-3(-3x^2 + 5x + 5)) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(2sin(3\u03c0(3|x + 2| - 3) + 2)) - 1| 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 minimum points for f(x) = -2(2cos(2\u03c0(1|x + 2| + 1)) + 3) - 4 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) = -2cos(3\u03c0(-1|(3x^2 + 1x + 4) - 2| + 1) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "533", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|(-3|(1x^2 - 5x + 3) + 3|) + 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (3(-2e^(2(5x^2 + 3x + 1)) + 3) + 3)/(2(-2e^(2(5x^2 + 3x + 1)) + 3) - 1) 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) = (-2(-5(3cos(3\u03c0x - 1) + 1)^2 - 5(3cos(3\u03c0x - 1) + 1) - 1) + 1)/(3(-5(3cos(3\u03c0x - 1) + 1)^2 - 5(3cos(3\u03c0x - 1) + 1) - 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 total number of local minima and maxima for f(x) = -3((-1(-2cos(1\u03c0x + 1) + 3) - 3)/(-1(-2cos(1\u03c0x + 1) + 3))) - 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 maxima for the function f(x) = 1(5(-3cos(2\u03c0x - 2) - 2)^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 number of local minimum points for f(x) = 5(-1cos(2\u03c0(4x - 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": "Determine the total count of local extrema points for the function f(x) = 3|(2cos(3\u03c0(3x - 5) + 1) - 3) - 2| + 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 maxima does the function f(x) = 2(1cos(1\u03c0((-1x + 3)/(2x - 1)) - 1) + 3)^2 + 1(1cos(1\u03c0((-1x + 3)/(2x - 1)) - 1) + 3) + 1 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) = 2sin(3\u03c0(-2|(4x + 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": "479", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(3\u03c0(-4(4x^2 - 1x - 2)^2 - 3(4x^2 - 1x - 2) + 5) + 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (1(1cos(3\u03c0((-2x - 1)/(2x - 1)) - 1) + 2) - 2)/(1(1cos(3\u03c0((-2x - 1)/(2x - 1)) - 1) + 2) + 1) 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) = 4(1cos(3\u03c0((2x - 3)/(-3x + 4)) - 3) + 3) + 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) = -2((3(1sin(3\u03c0x) + 2) - 2)/(-1(1sin(3\u03c0x) + 2)))^2 + 4((3(1sin(3\u03c0x) + 2) - 2)/(-1(1sin(3\u03c0x) + 2))) - 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 extrema (minima and maxima combined) does the function f(x) = (3(3cos(3\u03c0(2|x + 2| + 3) + 2) - 2) + 1)/(-2(3cos(3\u03c0(2|x + 2| + 3) + 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": "How many local maxima does the function f(x) = (-1(2(-1e^(-2x) - 2)^2 - 2(-1e^(-2x) - 2) - 4) - 1)/(-2(2(-1e^(-2x) - 2)^2 - 2(-1e^(-2x) - 2) - 4) - 3) 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": "Find the total number of local minima and maxima for f(x) = -4(-1|(-2sin(1\u03c0x - 2) - 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": "Determine the total count of local extrema points for the function f(x) = 4(-1sin(1\u03c0((1x - 2)/(-3x - 2)) + 1))^2 + 4(-1sin(1\u03c0((1x - 2)/(-3x - 2)) + 1)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(3cos(2\u03c0((-3x - 2)/(1x)) + 3) + 1)^2 + 5(3cos(2\u03c0((-3x - 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": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2\u03c0(-4(4x + 1)^2 - 4(4x + 1) - 1) - 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "417", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2cos(2\u03c0(1((-2x - 3)/(-1x + 3)))) + 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": "How many local minima does the function f(x) = (1(3cos(2\u03c0(2|x - 2| - 2)) - 1) + 1)/(-3(3cos(2\u03c0(2|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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3sin(3\u03c0(2(5x + 5)^2 + 4(5x + 5) - 5) - 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "190", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1|(1sin(1\u03c0(-3|x - 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": "32", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(1\u03c0(-2|(-3|x + 1| + 3) - 1|) + 3) - 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) = -1cos(1\u03c0(2(-5x + 4)^2 + 4(-5x + 4) - 3) - 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(5(-3|x + 3| + 1)^2 - 4(-3|x + 3| + 1) - 5)^2 - 3(5(-3|x + 3| + 1)^2 - 4(-3|x + 3| + 1) - 5) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3sin(2\u03c0(1|((3x - 3)/(-2x - 1)) - 2|) + 3) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 5(3e^(1(-2x^2 - 4x - 1)) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(-1sin(3\u03c0(-1x - 5) + 3) - 3)^2 + 2(-1sin(3\u03c0(-1x - 5) + 3) - 3) + 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 total count of local extrema points for the function f(x) = 1|((-1(1|x - 2| - 3))/(3(1|x - 2| - 3) + 1)) + 3| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (1(1(2sin(1\u03c0x - 3) + 2)^2 + 3(2sin(1\u03c0x - 3) + 2) + 5) + 3)/(2(1(2sin(1\u03c0x - 3) + 2)^2 + 3(2sin(1\u03c0x - 3) + 2) + 5) + 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 minimum points for f(x) = 3cos(2\u03c0(-3(-2|x + 1| + 3) - 5)) 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 minima for the function f(x) = 2sin(2\u03c0((-1((2x + 2)/(-1x + 3)))/(-2((2x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3((1(1sin(2\u03c0x) - 2) + 1)/(-3(1sin(2\u03c0x) - 2) + 2))^2 - 3((1(1sin(2\u03c0x) - 2) + 1)/(-3(1sin(2\u03c0x) - 2) + 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 minima for the function f(x) = 2sin(3\u03c0((-3(5x) + 1)/(2(5x))) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2|(-3(2cos(2\u03c0x - 1) + 2)^2 - 4(2cos(2\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3sin(3\u03c0(-3|(-3|x - 2| - 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": "269", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|((-1(-3cos(1\u03c0x + 3) - 1) - 1)/(-2(-3cos(1\u03c0x + 3) - 1) + 3)) + 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) = -1cos(3\u03c0(1(3x^2 + 2)^2 - 5(3x^2 + 2) + 3) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "113", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(-1(3sin(2\u03c0x - 3) - 2) - 5))/(2(-1(3sin(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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1(2cos(3\u03c0(-2|x + 1|) - 1) + 3)^2 - 4(2cos(3\u03c0(-2|x + 1|) - 1) + 3) + 3 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) = -5(3cos(3\u03c0(-4x + 3) + 2) - 1) + 4 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 minima does the function f(x) = -3cos(2\u03c0(1(-1|x + 1| + 1) + 5) - 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 minima does the function f(x) = 5(-5(1cos(2\u03c0x - 3) - 2) + 2)^2 - 5(-5(1cos(2\u03c0x - 3) - 2) + 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 total count of local extrema points for the function f(x) = (1(2cos(3\u03c0(2|x - 1| + 1) + 2)) - 1)/(-1(2cos(3\u03c0(2|x - 1| + 1) + 2)) - 1) 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 extrema (minima and maxima combined) does the function f(x) = 5(-2cos(2\u03c0((-3x + 3)/(3x + 2)) - 1) + 2)^2 + 2(-2cos(2\u03c0((-3x + 3)/(3x + 2)) - 1) + 2) + 2 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 minima does the function f(x) = 1(3(2cos(1\u03c0x + 1) - 2)^2 - 3(2cos(1\u03c0x + 1) - 2) + 2) - 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 minima does the function f(x) = 2(2cos(1\u03c0(-3x^2 - 5x - 4) - 2) + 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "279", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2((1(3sin(2\u03c0x + 3) + 2))/(1(3sin(2\u03c0x + 3) + 2) - 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) = 1|(2|(-2cos(3\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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1|(-1(3sin(2\u03c0x + 1) + 2)^2 - 1) - 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 extrema (minima and maxima combined) does the function f(x) = -2sin(3\u03c0(5(-3|x - 1| - 3)^2 + 2(-3|x - 1| - 3)) - 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "403", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-2((-2(-3cos(2\u03c0x + 3) - 1))/(-3(-3cos(2\u03c0x + 3) - 1))) + 1)/(-1((-2(-3cos(2\u03c0x + 3) - 1))/(-3(-3cos(2\u03c0x + 3) - 1))) + 2) have 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) = -5((-2(-3cos(3\u03c0x + 2) - 1) + 2)/(-1(-3cos(3\u03c0x + 2) - 1) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1sin(1\u03c0(1|(-1x^2 - 4x + 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": "211", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5(1(-4x^2 - 2x + 1)^2 - 4)^2 + 4(1(-4x^2 - 2x + 1)^2 - 4) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(-2sin(2\u03c0((-1x - 3)/(-3x - 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": "Find the number of local minimum points for f(x) = -3sin(3\u03c0((-3((3x - 2)/(-3x)) - 3)/(-3((3x - 2)/(-3x)) - 2)) + 2) + 1 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": "How many local minima does the function f(x) = 1cos(2\u03c0(-3(2|x + 3|) - 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": "Determine the total count of local extrema points for the function f(x) = 2(2cos(3\u03c0(1x + 3) + 3) - 3)^2 + 2(2cos(3\u03c0(1x + 3) + 3) - 3) - 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(5(2cos(3\u03c0x - 3) + 3)^2 - 4(2cos(3\u03c0x - 3) + 3) + 1)^2 - 1(5(2cos(3\u03c0x - 3) + 3)^2 - 4(2cos(3\u03c0x - 3) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3cos(3\u03c0((-3(1x^2 - 3x - 1) - 3)/(-2(1x^2 - 3x - 1) - 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": "Determine the count of local maxima for the function f(x) = 3|(-2cos(1\u03c0((1x)/(-2x - 1)) - 3) + 3) + 2| + 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) = -1cos(2\u03c0(-5((-1x - 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1|((2(-2|x + 2| + 1) - 3)/(2(-2|x + 2| + 1) - 1)) + 2| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(-1sin(1\u03c0((-1x - 3)/(-1x)) - 1) - 3)^2 + 5(-1sin(1\u03c0((-1x - 3)/(-1x)) - 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) = -1cos(2\u03c0((2((-3x + 1)/(3x)) + 2)/(-3((-3x + 1)/(3x)) + 1)) - 1) + 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 count of local maxima for the function f(x) = 1(-1(2sin(2\u03c0x + 3))^2 - 3(2sin(2\u03c0x + 3)) + 1) - 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 maxima does the function f(x) = -2sin(3\u03c0((-2((1x - 3)/(1x + 3)) - 3)/(1((1x - 3)/(1x + 3))))) + 2 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) = 2|(-1(1sin(2\u03c0x - 3) + 3)^2 + 4(1sin(2\u03c0x - 3) + 3) - 1) + 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": "Find the total number of local minima and maxima for f(x) = -2|(4(-1|x - 1| + 2)^2 - 5(-1|x - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (1(1sin(3\u03c0(3|x + 2| + 3) + 2) + 3) + 3)/(1(1sin(3\u03c0(3|x + 2| + 3) + 2) + 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2|(1cos(2\u03c0(-4x^2 - 3x - 5) + 2) - 1) - 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": "Determine the total count of local extrema points for the function f(x) = -3|(-3sin(1\u03c0(2x^2 - 4x - 1) - 1) + 2) - 3| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "532", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2|(-3e^(2((-3x + 3)/(1x + 3))) - 2) + 2| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-2((2(1x^2 + 2x + 3) + 2)/(-3(1x^2 + 2x + 3) - 3)))/(-2((2(1x^2 + 2x + 3) + 2)/(-3(1x^2 + 2x + 3) - 3)) + 3) 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": "How many local extrema (minima and maxima combined) does the function f(x) = -1(1cos(3\u03c0(-3x + 1) - 1) - 2)^2 + 4(1cos(3\u03c0(-3x + 1) - 1) - 2) + 4 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": "Determine the count of local maxima for the function f(x) = -5(-1sin(2\u03c0(1x + 2)))^2 - 5(-1sin(2\u03c0(1x + 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 maxima does the function f(x) = -1(-2(1cos(1\u03c0x + 2) - 2)^2 - 2(1cos(1\u03c0x + 2) - 2) + 5)^2 + 4(-2(1cos(1\u03c0x + 2) - 2)^2 - 2(1cos(1\u03c0x + 2) - 2) + 5) - 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": "Find the number of local minimum points for f(x) = (-2((1(2cos(1\u03c0x - 1) + 2) - 2)/(-1(2cos(1\u03c0x - 1) + 2) - 3)) - 2)/(-3((1(2cos(1\u03c0x - 1) + 2) - 2)/(-1(2cos(1\u03c0x - 1) + 2) - 3)) - 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 number of local minimum points for f(x) = -2sin(3\u03c0(3|(1|x - 2| + 3) - 2| + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 5(-2cos(1\u03c0(2|x - 3| + 1) - 3) + 1) + 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": "Find the total number of local minima and maxima for f(x) = -1|(1|(-1cos(2\u03c0x + 3) - 2) + 2| + 3) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2cos(3\u03c0(-1|((-1x - 1)/(-2x + 1)) + 1| + 1)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4(2|(2cos(1\u03c0x + 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 extrema (minima and maxima combined) does the function f(x) = 2|(1sin(3\u03c0(2x) + 3) + 1) - 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": "How many local minima does the function f(x) = -1((-1(2sin(2\u03c0x) - 1) + 1)/(3(2sin(2\u03c0x) - 1) + 3))^2 + 3((-1(2sin(2\u03c0x) - 1) + 1)/(3(2sin(2\u03c0x) - 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) = 3cos(2\u03c0(1|(2|x - 2| - 3) - 1| - 3) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2(-1(1|x - 2| - 3)^2 + 5(1|x - 2| - 3) + 4)^2 + 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(3sin(2\u03c0x + 2) + 3)^2 + 5(3sin(2\u03c0x + 2) + 3) + 4) - 1)/(2(-1(3sin(2\u03c0x + 2) + 3)^2 + 5(3sin(2\u03c0x + 2) + 3) + 4) - 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 minima for the function f(x) = (3(-4(1cos(3\u03c0x + 1)) + 1) - 1)/(1(-4(1cos(3\u03c0x + 1)) + 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": "How many local maxima does the function f(x) = 1|(-1(3sin(1\u03c0x + 1))^2 + 4(3sin(1\u03c0x + 1)) - 5) + 3| - 3 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": "Determine the count of local maxima for the function f(x) = (2(-3cos(1\u03c0(-5x + 1) - 2) + 2) + 2)/(2(-3cos(1\u03c0(-5x + 1) - 2) + 2) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(-1sin(1\u03c0(5x - 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": "How many local minima does the function f(x) = -1(1|(2sin(1\u03c0x + 3)) + 2| - 2) + 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": "Determine the count of local minima for the function f(x) = 4(2|(1cos(1\u03c0x - 1) + 2) + 3|)^2 + 5(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": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -4(3(-1sin(3\u03c0x - 2) - 2) - 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 total count of local extrema points for the function f(x) = 3(-1sin(2\u03c0(2x - 1) + 3) - 3) - 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) = -3|((-1(2sin(3\u03c0x + 3) + 2) - 3)/(-1(2sin(3\u03c0x + 3) + 2) + 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 total count of local extrema points for the function f(x) = 5(1(3cos(1\u03c0x) - 3)^2 - 5(3cos(1\u03c0x) - 3) + 5)^2 - 3(1(3cos(1\u03c0x) - 3)^2 - 5(3cos(1\u03c0x) - 3) + 5) + 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 minima does the function f(x) = -3(-3cos(1\u03c0(-3x^2 + 1x + 1) - 3))^2 + 2(-3cos(1\u03c0(-3x^2 + 1x + 1) - 3)) - 2 have 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 extrema (minima and maxima combined) does the function f(x) = -1|(1(-3cos(3\u03c0x - 2) - 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": "Find the number of local maximum points for f(x) = (-2(-2cos(1\u03c0(-3|x - 2| + 1) - 1) + 3) - 2)/(3(-2cos(1\u03c0(-3|x - 2| + 1) - 1) + 3) + 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": "Determine the count of local minima for the function f(x) = -2cos(3\u03c0(-4(1x + 1)^2 - 5(1x + 1) - 2) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "251", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(3sin(1\u03c0((-2x + 3)/(1x + 2)) - 1) + 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(2sin(1\u03c0(-2x + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(2(-1cos(3\u03c0x - 3) - 3)^2 - 2) - 5 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) = -1sin(3\u03c0(4(-3|x - 3| - 1) + 3) - 2) + 2 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 maxima does the function f(x) = (3(1sin(1\u03c0(-4x^2 - 3x - 1) + 1) + 2) - 1)/(-2(1sin(1\u03c0(-4x^2 - 3x - 1) + 1) + 2) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "328", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-2(-5(-3sin(3\u03c0x + 2) - 3) + 2) + 1)/(3(-5(-3sin(3\u03c0x + 2) - 3) + 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": "Find the number of local maximum points for f(x) = -3|(-1sin(3\u03c0((-1x + 3)/(-3x + 4)) + 3) - 2) - 3| + 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": "Find the total number of local minima and maxima for f(x) = (-2(-1e^(-2((-3x - 3)/(-3x))) + 3) - 3)/(3(-1e^(-2((-3x - 3)/(-3x))) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1|((3(-1x^2 + 4x + 3) - 2)/(-2(-1x^2 + 4x + 3))) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -5((-1(3sin(3\u03c0x) - 2) + 3)/(-3(3sin(3\u03c0x) - 2) - 1)) - 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": "Find the total number of local minima and maxima for f(x) = -2|((-1(-2sin(3\u03c0x + 1) + 3) - 3)/(-3(-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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3|(3cos(2\u03c0((-2x - 2)/(-3x - 2)) - 3) - 3) - 2| - 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 extrema (minima and maxima combined) does the function f(x) = 3cos(3\u03c0(5((-1x - 1)/(3x - 1))^2 - 3((-1x - 1)/(3x - 1)) + 3)) + 2 have in the interval [-10, 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) = -1sin(1\u03c0(1(3x + 1) + 4) + 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 minima does the function f(x) = 1(3cos(3\u03c0(-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": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 2sin(3\u03c0(1(2x - 2)^2 - 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": "253", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1sin(2\u03c0(-1(2x + 3)^2 + 4(2x + 3) - 5)) 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 count of local maxima for the function f(x) = -2cos(3\u03c0(5(-1|x + 1| + 2) - 1) + 1) - 2 in the interval [-10, 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) = 1cos(2\u03c0((-2((2x + 2)/(-2x)) + 1)/(1((2x + 2)/(-2x)) + 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": "Find the total number of local minima and maxima for f(x) = 1cos(1\u03c0(-2|((3x + 2)/(-1x)) + 1| - 2) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1e^(1(5(-3x^2 + 4)^2 + 4(-3x^2 + 4) + 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 5((1(-3sin(1\u03c0x - 2) - 3) - 2)/(-2(-3sin(1\u03c0x - 2) - 3) - 1))^2 + 3((1(-3sin(1\u03c0x - 2) - 3) - 2)/(-2(-3sin(1\u03c0x - 2) - 3) - 1)) - 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 maxima for the function f(x) = 1cos(2\u03c0(2|(3|x + 2| - 1) + 2| + 1) + 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": "How many local maxima does the function f(x) = 2cos(3\u03c0(2|(4x + 1) - 2| - 2) - 3) have 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(1\u03c0x + 2) + 2) + 2|) + 1| - 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 extrema (minima and maxima combined) does the function f(x) = -3(2sin(1\u03c0((1x + 2)/(-1x - 3)) + 2) - 1)^2 + 2(2sin(1\u03c0((1x + 2)/(-1x - 3)) + 2) - 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": "Determine the total count of local extrema points for the function f(x) = -2cos(1\u03c0(-4(-1|x + 2| - 2) - 5) - 1) + 3 in the interval [-10, 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) = -1sin(2\u03c0(1|((2x - 2)/(-3x)) + 3| + 3) + 1) + 2 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 minima for the function f(x) = -2|(-2(-1sin(1\u03c0x - 2))^2 + 5(-1sin(1\u03c0x - 2)) - 5) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3(-3cos(1\u03c0(-3|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": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1(2(2cos(1\u03c0x - 1) + 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": "How many local minima does the function f(x) = 1(2cos(2\u03c0(3x^2 - 3x - 1)) - 1)^2 + 3(2cos(2\u03c0(3x^2 - 3x - 1)) - 1) + 4 have in the interval [-10, 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) = (-1((2(3sin(3\u03c0x - 1) + 1))/(2(3sin(3\u03c0x - 1) + 1) + 2)) + 2)/(-3((2(3sin(3\u03c0x - 1) + 1))/(2(3sin(3\u03c0x - 1) + 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 minima for the function f(x) = -1sin(2\u03c0(-2|(5x + 1) - 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": "Find the number of local maximum points for f(x) = -3(5(1x^2 - 4x - 4)^2 + 4(1x^2 - 4x - 4))^2 - 1(5(1x^2 - 4x - 4)^2 + 4(1x^2 - 4x - 4)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(-4(-2cos(2\u03c0x + 3) - 1)^2 - 5(-2cos(2\u03c0x + 3) - 1) + 2) - 3 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) = (1(1(2cos(3\u03c0x - 3) + 2) - 1) - 2)/(2(1(2cos(3\u03c0x - 3) + 2) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = (-3(1sin(2\u03c0(-1|x - 3| + 2) - 2)))/(2(1sin(2\u03c0(-1|x - 3| + 2) - 2)) + 1) 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 number of local maximum points for f(x) = 1(2sin(1\u03c0(-4x + 1) - 2) - 1)^2 - 1(2sin(1\u03c0(-4x + 1) - 2) - 1) - 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(3sin(2\u03c0((1x + 3)/(1x - 2)) + 3) - 2) - 2)/(1(3sin(2\u03c0((1x + 3)/(1x - 2)) + 3) - 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": "How many local maxima does the function f(x) = -2e^(-1(-3(2x - 1) - 2)) + 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": "Determine the count of local maxima for the function f(x) = 2sin(2\u03c0((2(1x^2 + 2x - 1) + 1)/(-2(1x^2 + 2x - 1) + 3)) + 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 count of local minima for the function f(x) = -4(-1|(-1cos(3\u03c0x + 2)) + 3| - 3)^2 - 4(-1|(-1cos(3\u03c0x + 2)) + 3| - 3) - 4 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) = 2|((3(-1sin(3\u03c0x + 2) - 3) + 1)/(-2(-1sin(3\u03c0x + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 2(2sin(1\u03c0(-3x + 4) - 3) + 1)^2 - 4 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 maximum points for f(x) = 3|((-2(1x^2 + 5x - 5) - 1)/(-3(1x^2 + 5x - 5) + 1)) + 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-3((-3(-2cos(1\u03c0x - 1) - 3))/(-2(-2cos(1\u03c0x - 1) - 3))) + 3)/(-3((-3(-2cos(1\u03c0x - 1) - 3))/(-2(-2cos(1\u03c0x - 1) - 3))) - 2) 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 number of local minimum points for f(x) = 3(5(-2sin(2\u03c0x + 3) + 2) + 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) = 2cos(2\u03c0(-3(5x - 3)^2 + 1(5x - 3) + 4) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "208", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(-1cos(1\u03c0(-1|x| + 3) + 2) - 3) - 2| - 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": "Determine the count of local minima for the function f(x) = (1(-5(1sin(2\u03c0x + 3) + 3)^2 + 1(1sin(2\u03c0x + 3) + 3) + 5) + 2)/(-2(-5(1sin(2\u03c0x + 3) + 3)^2 + 1(1sin(2\u03c0x + 3) + 3) + 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": "Determine the total count of local extrema points for the function f(x) = 3cos(3\u03c0(-1(-5x^2 + 1x + 3) + 5) + 2) + 1 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) = -1cos(2\u03c0(3(5x - 2)^2 - 4(5x - 2) - 1) - 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "426", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2(1|(-3cos(1\u03c0x + 2) - 2) + 3|)^2 + 3(1|(-3cos(1\u03c0x + 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": "How many local maxima does the function f(x) = -1|(-3|(-2cos(1\u03c0x - 3) + 3) + 2| - 2) - 1| - 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 minima does the function f(x) = 2cos(1\u03c0(-1((2x + 1)/(-2x - 2))^2 - 4((2x + 1)/(-2x - 2)) - 1) + 2) + 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": "Find the total number of local minima and maxima for f(x) = -2sin(3\u03c0(-2(1x^2 - 5x - 1) - 3) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "519", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3cos(1\u03c0(1(1x)^2 - 2(1x)) - 3) - 3 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) = 1(3sin(1\u03c0(5x + 3) - 1) - 1)^2 - 2(3sin(1\u03c0(5x + 3) - 1) - 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) = 1|(4(-2|x - 2| + 2)^2 - 4(-2|x - 2| + 2) - 5) - 2| + 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 maximum points for f(x) = (-2(2(3cos(3\u03c0x - 1) - 1)^2 + 5(3cos(3\u03c0x - 1) - 1) - 4))/(-2(2(3cos(3\u03c0x - 1) - 1)^2 + 5(3cos(3\u03c0x - 1) - 1) - 4) - 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 minima for the function f(x) = 3cos(3\u03c0(5(-2x^2 + 1x + 2) + 3) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "237", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(-2sin(3\u03c0((2x - 3)/(2x + 1)) + 3) + 3) + 4 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) = 3sin(3\u03c0(-2|(3|x - 2| - 3) + 2|)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "363", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1(2sin(1\u03c0(-1|x + 2|) - 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": "Determine the count of local maxima for the function f(x) = -5(-2cos(1\u03c0(3x + 1) + 3) - 2)^2 - 3(-2cos(1\u03c0(3x + 1) + 3) - 2) - 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": "Find the total number of local minima and maxima for f(x) = (1(-3cos(3\u03c0(2|x| - 1) - 3) + 2) - 3)/(-1(-3cos(3\u03c0(2|x| - 1) - 3) + 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 total count of local extrema points for the function f(x) = 3sin(1\u03c0(1(-3|x - 1| + 3) - 3)) + 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 extrema (minima and maxima combined) does the function f(x) = 2cos(1\u03c0(1(-4x + 3) - 3)) + 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) = -3sin(1\u03c0(1|(3|x + 1| - 2) + 2| + 2) - 1) - 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": "Determine the count of local minima for the function f(x) = (3(-1e^(-2(-5x^2 - 4)) - 1) - 1)/(-1(-1e^(-2(-5x^2 - 4)) - 1) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3|(-2(-3sin(3\u03c0x - 3) + 2) - 2) - 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": "Find the total number of local minima and maxima for f(x) = 3|(-2sin(1\u03c0(-3x^2 + 4x + 5) - 2) + 3) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "564", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -4(-1cos(3\u03c0(-1x^2 - 3x - 4) - 1) - 1) - 4 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 extrema (minima and maxima combined) does the function f(x) = 3cos(2\u03c0(-5(-1x^2 + 4x + 1))) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "500", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (2(2sin(3\u03c0(-1|x - 2| + 3) + 3) + 2) - 1)/(-3(2sin(3\u03c0(-1|x - 2| + 3) + 3) + 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": "How many local minima does the function f(x) = 2sin(3\u03c0(-2((-1x + 1)/(-1x - 1)) - 2) - 3) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 5(2sin(1\u03c0((1x)/(-3x + 2))) + 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": "How many local maxima does the function f(x) = -2cos(3\u03c0(-3|(1x^2 - 1x + 5) + 1| - 3) - 3) + 2 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) = 1|(-1cos(1\u03c0(-4x^2 - 3x - 1) - 3) - 3) - 3| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "288", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1cos(2\u03c0(4(2|x - 1| + 2)^2 - 5(2|x - 1| + 2) - 4)) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3((1(-3sin(1\u03c0x + 1) - 3) - 3)/(1(-3sin(1\u03c0x + 1) - 3) + 1))^2 - 2((1(-3sin(1\u03c0x + 1) - 3) - 3)/(1(-3sin(1\u03c0x + 1) - 3) + 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": "Find the number of local minimum points for f(x) = -1cos(3\u03c0((-1(-4x - 3) - 1)/(-1(-4x - 3) - 3)) + 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 maximum points for f(x) = -1|(3cos(1\u03c0(-3|x + 2| - 3) - 2) - 3) - 3| + 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": "How many local minima does the function f(x) = -1sin(1\u03c0(3(2|x - 3| - 1)^2 - 5(2|x - 3| - 1) - 5) - 3) - 3 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 total number of local minima and maxima for f(x) = 2e^(-1(-3|(2x^2 - 5x - 2) - 2| - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -5(-3sin(2\u03c0(-4x - 2) - 2) - 1)^2 + 2(-3sin(2\u03c0(-4x - 2) - 2) - 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1cos(2\u03c0((-1(-3x + 4) + 1)/(1(-3x + 4))) - 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) = -3sin(3\u03c0(-1(-3|x + 3| + 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": "Find the total number of local minima and maxima for f(x) = (2(-1sin(2\u03c0(-3x - 3) - 2) - 3))/(1(-1sin(2\u03c0(-3x - 3) - 2) - 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": "Determine the count of local maxima for the function f(x) = (-2(-2sin(2\u03c0(4x) + 3) - 2) - 3)/(3(-2sin(2\u03c0(4x) + 3) - 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": "Determine the count of local minima for the function f(x) = (-2(-1cos(1\u03c0(-2x^2 + 4x + 1) + 2) - 3))/(2(-1cos(1\u03c0(-2x^2 + 4x + 1) + 2) - 3) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(1(-1|x - 2| - 1)^2 + 4(-1|x - 2| - 1) + 1)^2 - 1(1(-1|x - 2| - 1)^2 + 4(-1|x - 2| - 1) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3|((3(1x^2 + 2x + 4) + 3)/(-1(1x^2 + 2x + 4) - 1)) + 3| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "139", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(4(-2sin(3\u03c0x - 2))^2 - 4(-2sin(3\u03c0x - 2)) - 1) - 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 minimum points for f(x) = 1(2cos(2\u03c0(-2|x - 1|) + 1) + 2)^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) = -1(3sin(2\u03c0(-1|x - 1| - 3) + 1)) - 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 extrema (minima and maxima combined) does the function f(x) = 1|(-3cos(1\u03c0(-4x - 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": "144", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|(-1sin(3\u03c0(2x^2 + 2x - 1) - 3) + 1) - 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "223", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1((-3(-2cos(1\u03c0x - 1)) + 1)/(-2(-2cos(1\u03c0x - 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 extrema (minima and maxima combined) does the function f(x) = 3sin(3\u03c0((-1(2x^2 + 5x + 2) + 3)/(-2(2x^2 + 5x + 2) + 3)) + 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": "How many local maxima does the function f(x) = (3(-3(1sin(1\u03c0x - 2) + 2)^2 + 1(1sin(1\u03c0x - 2) + 2) + 3) + 3)/(-2(-3(1sin(1\u03c0x - 2) + 2)^2 + 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": "Determine the count of local maxima for the function f(x) = -3sin(2\u03c0(1|((-3x - 1)/(3x)) - 1| + 3) - 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 total number of local minima and maxima for f(x) = (2(-1(-3sin(1\u03c0x + 3) - 1) + 1) - 1)/(-3(-1(-3sin(1\u03c0x + 3) - 1) + 1) + 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 total number of local minima and maxima for f(x) = -2cos(1\u03c0(-5(-1x^2 + 1) - 2) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "501", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1(-3sin(1\u03c0(3|x + 3| - 3)) - 2) - 5 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) = -2cos(2\u03c0(-3|(-2|x - 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|((-3(2sin(3\u03c0x) - 1) - 2)/(-2(2sin(3\u03c0x) - 1) + 3)) + 2| - 1 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 maximum points for f(x) = (3(2sin(1\u03c0(-3|x + 2| + 2) - 3)) - 3)/(-1(2sin(1\u03c0(-3|x + 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": "Find the number of local minimum points for f(x) = -1|(1cos(3\u03c0((-2x + 1)/(-3x - 1)) - 1) - 3) + 1| + 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 extrema (minima and maxima combined) does the function f(x) = 2(-3e^(-2(1|x + 2| - 1)) + 1)^2 - 4(-3e^(-2(1|x + 2| - 1)) + 1) + 3 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(1cos(3\u03c0(-4x - 2) + 1) + 2)^2 + 1(1cos(3\u03c0(-4x - 2) + 1) + 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": "How many local minima does the function f(x) = (-1(1sin(3\u03c0((2x - 2)/(-3x - 2)))) - 2)/(-1(1sin(3\u03c0((2x - 2)/(-3x - 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 total count of local extrema points for the function f(x) = 3(-2cos(1\u03c0(-2x^2 + 5x - 1) - 2) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "407", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -1(-3cos(2\u03c0(2|x - 2| + 3))) - 4 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 count of local maxima for the function f(x) = -1cos(2\u03c0(-2|(-4x) - 1| - 1) + 2) + 2 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 number of local minimum points for f(x) = (1(-2sin(3\u03c0(2x^2 + 3x + 4) - 2) - 3) + 1)/(-1(-2sin(3\u03c0(2x^2 + 3x + 4) - 2) - 3) + 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": "Find the number of local minimum points for f(x) = -1cos(1\u03c0(-1|((2x)/(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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2cos(2\u03c0(1(-1x^2 + 5x)^2 + 1(-1x^2 + 5x) + 2) - 3) - 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 total count of local extrema points for the function f(x) = -2(-2cos(3\u03c0(-1|x|) - 2) + 1) + 5 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) = 3(-3e^(2(-3|x + 3| + 3)) - 3)^2 + 1(-3e^(2(-3|x + 3| + 3)) - 3) + 2 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 minima does the function f(x) = -5(1(-2sin(3\u03c0x + 3) - 1) + 3)^2 - 1(1(-2sin(3\u03c0x + 3) - 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 total number of local minima and maxima for f(x) = 2|(1|(1cos(2\u03c0x - 3) - 2)| - 1) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2((-1(-1cos(3\u03c0x + 2) - 3) + 1)/(-3(-1cos(3\u03c0x + 2) - 3)))^2 + 2((-1(-1cos(3\u03c0x + 2) - 3) + 1)/(-3(-1cos(3\u03c0x + 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": "How many local maxima does the function f(x) = 1|(-2(3sin(1\u03c0x - 1) + 2)^2 + 3(3sin(1\u03c0x - 1) + 2) + 2)| - 1 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": "Determine the count of local maxima for the function f(x) = 2|(-1sin(3\u03c0(3|x - 1| + 2) + 1) - 1) + 2| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -5(-3(-3sin(2\u03c0x) - 2)^2 + 2(-3sin(2\u03c0x) - 2) - 4)^2 - 1(-3(-3sin(2\u03c0x) - 2)^2 + 2(-3sin(2\u03c0x) - 2) - 4) - 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 minima does the function f(x) = 4(1|(-1sin(3\u03c0x + 3) + 3) - 1| - 1) + 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 total count of local extrema points for the function f(x) = -2sin(2\u03c0(-3(-4x^2 + 4x - 3) - 3) - 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "501", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2cos(3\u03c0(3(1|x + 2| + 2)^2 + 1(1|x + 2| + 2) - 3) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "545", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2(-3sin(1\u03c0((-2x - 1)/(1x + 2)) + 3) - 3) + 5 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) = -1cos(2\u03c0(-1(1x^2 - 2x - 2)^2 - 2(1x^2 - 2x - 2) - 3) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2cos(3\u03c0(4(4x^2) + 1) - 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "232", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3|(2|(1sin(3\u03c0x + 2))|) + 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "105", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3sin(2\u03c0(-3(-4x - 2)^2 + 5(-4x - 2) + 3) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "428", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2(1(2cos(1\u03c0x) + 2) + 2) - 4 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 minima does the function f(x) = -5(-1sin(2\u03c0(-2x) + 1) + 3)^2 + 4(-1sin(2\u03c0(-2x) + 1) + 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": "Find the total number of local minima and maxima for f(x) = -3(-3sin(2\u03c0((-3x - 2)/(-1x))) + 2) + 1 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 count of local maxima for the function f(x) = (2(1e^(2(-2|x + 1| - 2))))/(-3(1e^(2(-2|x + 1| - 2)))) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "112", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 5((3(3x^2 - 4) - 2)/(2(3x^2 - 4) + 3))^2 - 4((3(3x^2 - 4) - 2)/(2(3x^2 - 4) + 3)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-3((-1(2cos(1\u03c0x - 2) + 2) + 2)/(2(2cos(1\u03c0x - 2) + 2) + 2)) + 3)/(-2((-1(2cos(1\u03c0x - 2) + 2) + 2)/(2(2cos(1\u03c0x - 2) + 2) + 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 maxima does the function f(x) = (-2(-1e^(-2(5x - 1))) - 3)/(3(-1e^(-2(5x - 1)))) 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": "How many local maxima does the function f(x) = 1(2|(2sin(3\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": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3|(-1sin(1\u03c0(-2|x + 3| + 1) + 3) - 2) - 3| + 2 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((-1(1x^2 + 4x - 1) + 3)/(-1(1x^2 + 4x - 1) + 2))^2 + 5((-1(1x^2 + 4x - 1) + 3)/(-1(1x^2 + 4x - 1) + 2)) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1(-2(2sin(1\u03c0x - 2))^2 - 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": "Find the total number of local minima and maxima for f(x) = (1(2cos(3\u03c0(-5x - 5)) + 3))/(3(2cos(3\u03c0(-5x - 5)) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "295", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3|(-1(-2cos(1\u03c0x - 2) + 3) + 4) - 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 total number of local minima and maxima for f(x) = 3(-1cos(3\u03c0((-2x - 1)/(1x - 3)) + 1) + 3) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2|(-2sin(3\u03c0(5x - 2) + 2)) + 3| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "150", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4(-1|(-2sin(3\u03c0x + 3) - 3) - 3| + 2)^2 - 2(-1|(-2sin(3\u03c0x + 3) - 3) - 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 minima does the function f(x) = -1cos(3\u03c0(-2|(3x + 4) - 1|) - 3) + 1 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": "Determine the count of local minima for the function f(x) = 1sin(2\u03c0((1(-2x - 1) - 1)/(-1(-2x - 1) + 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": "Determine the count of local minima for the function f(x) = 2sin(3\u03c0((-2((2x)/(2x + 1)) + 3)/(2((2x)/(2x + 1))))) - 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": "Find the total number of local minima and maxima for f(x) = 3((1(3cos(1\u03c0x + 3) - 2) - 1)/(1(3cos(1\u03c0x + 3) - 2) - 2))^2 - 4((1(3cos(1\u03c0x + 3) - 2) - 1)/(1(3cos(1\u03c0x + 3) - 2) - 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": "Find the number of local minimum points for f(x) = -2|(-1(-2cos(3\u03c0x + 2) - 1)^2 + 4(-2cos(3\u03c0x + 2) - 1) - 5) + 3| + 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": "Determine the count of local maxima for the function f(x) = -1(-1e^(1(-5x - 2)) - 2) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(3sin(2\u03c0(1|x + 2| - 2) - 3) - 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": "Determine the count of local minima for the function f(x) = 1((2(5x^2 - 1x - 2) - 3)/(2(5x^2 - 1x - 2) + 2))^2 + 2((2(5x^2 - 1x - 2) - 3)/(2(5x^2 - 1x - 2) + 2)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -5(4(3cos(3\u03c0x) - 2) + 5)^2 - 1(4(3cos(3\u03c0x) - 2) + 5) + 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) = (-2(2(1cos(3\u03c0x + 2) - 3)^2 + 2(1cos(3\u03c0x + 2) - 3) + 4) - 2)/(-2(2(1cos(3\u03c0x + 2) - 3)^2 + 2(1cos(3\u03c0x + 2) - 3) + 4) + 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) = 3sin(3\u03c0((1(-1x^2 + 4x + 2) - 2)/(2(-1x^2 + 4x + 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) = 1(-2sin(2\u03c0(-4x^2 + 3x - 1) - 1))^2 + 3(-2sin(2\u03c0(-4x^2 + 3x - 1) - 1)) - 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "291", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2|(-1e^(-1(3x - 3)) - 3) + 2| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3cos(1\u03c0(-2((-1x + 2)/(2x + 1))^2 - 2((-1x + 2)/(2x + 1))) - 1) - 2 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": "Determine the total count of local extrema points for the function f(x) = -1(-3sin(3\u03c0((2x - 1)/(3x - 1)) - 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": "Find the number of local minimum points for f(x) = -3|(4(-1x^2 - 4x + 3)^2 + 2(-1x^2 - 4x + 3) - 3) - 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": "Determine the total count of local extrema points for the function f(x) = 2(1cos(1\u03c0(2|x|) + 2) + 2) - 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": "Determine the count of local maxima for the function f(x) = -4((2(-2sin(3\u03c0x) + 3))/(-2(-2sin(3\u03c0x) + 3) + 3)) - 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 maxima does the function f(x) = -5((1(3cos(2\u03c0x + 1) + 3) - 3)/(2(3cos(2\u03c0x + 1) + 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) = -2|(1cos(3\u03c0(-4x + 5)) + 2) - 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 minima does the function f(x) = (2((-2(3x^2 + 3x - 3) - 2)/(-3(3x^2 + 3x - 3) - 3)) + 1)/(2((-2(3x^2 + 3x - 3) - 2)/(-3(3x^2 + 3x - 3) - 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": "Find the number of local minimum points for f(x) = 1|(-1sin(1\u03c0(-3x - 5) - 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 minimum points for f(x) = 2cos(2\u03c0(1(5x + 4) - 1) + 3) + 3 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": "How many local maxima does the function f(x) = 3(2cos(1\u03c0(2x^2 + 4x + 2) - 2) - 3)^2 + 2(2cos(1\u03c0(2x^2 + 4x + 2) - 2) - 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "189", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3cos(1\u03c0(1(-1|x + 1| - 2)^2 - 1(-1|x + 1| - 2) + 2) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "303", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3sin(3\u03c0(1(-5x)^2 - 1(-5x) - 5) + 1) - 1 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": "Determine the total count of local extrema points for the function f(x) = 1cos(1\u03c0(2|(3|x - 3| + 2) - 3| + 3) + 3) + 1 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 minima does the function f(x) = (-1(-2sin(3\u03c0(-4x^2 - 1x - 5) + 3) + 3) - 2)/(-1(-2sin(3\u03c0(-4x^2 - 1x - 5) + 3) + 3) - 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "331", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (2(2sin(3\u03c0((1x + 2)/(-1x)) + 2) + 3) - 1)/(-2(2sin(3\u03c0((1x + 2)/(-1x)) + 2) + 3) + 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 extrema (minima and maxima combined) does the function f(x) = 2(-2sin(1\u03c0(-4x^2 + 3x + 2) + 1) - 3) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "573", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2(1sin(2\u03c0(-4x^2 - 4x - 2) + 2) - 2)^2 + 4(1sin(2\u03c0(-4x^2 - 4x - 2) + 2) - 2) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "211", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(1(-3sin(3\u03c0x - 3) - 2)^2 - 3(-3sin(3\u03c0x - 3) - 2)) + 2)/(-3(1(-3sin(3\u03c0x - 3) - 2)^2 - 3(-3sin(3\u03c0x - 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 minima for the function f(x) = 2(5(1sin(3\u03c0x) - 1)^2 + 4(1sin(3\u03c0x) - 1) + 1)^2 - 2(5(1sin(3\u03c0x) - 1)^2 + 4(1sin(3\u03c0x) - 1) + 1) + 5 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) = -3((-3(1x^2 - 1x - 3))/(-1(1x^2 - 1x - 3)))^2 + 5((-3(1x^2 - 1x - 3))/(-1(1x^2 - 1x - 3))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "176", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2sin(2\u03c0(4(4x^2 - 2x + 5)) + 2) - 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": "How many local maxima does the function f(x) = -3cos(1\u03c0(3|(-1|x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(3sin(2\u03c0(1x + 4)) - 1) - 2)/(-1(3sin(2\u03c0(1x + 4)) - 1) - 1) 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) = -2sin(3\u03c0(4(-4x + 3) + 1) + 1) + 2 in the interval [-10, 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|(2|(-1sin(2\u03c0x - 3)) - 1| - 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": "How many local maxima does the function f(x) = -2|((-2(-3cos(3\u03c0x + 1)) - 1)/(3(-3cos(3\u03c0x + 1)) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1sin(1\u03c0(-1(-2|x + 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3sin(3\u03c0(2|(-3|x| + 3) + 2| - 2) - 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": "How many local maxima does the function f(x) = 1cos(1\u03c0(1|(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": "10", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2sin(1\u03c0(4(-3x + 3) - 2) + 2) + 3 have 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 count of local minima for the function f(x) = -2cos(2\u03c0(3(1|x - 1|)^2 + 5(1|x - 1|) - 1) - 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "248", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1(2sin(2\u03c0((-3x + 2)/(-1x)) + 3) + 3)^2 + 2(2sin(2\u03c0((-3x + 2)/(-1x)) + 3) + 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": "Determine the count of local minima for the function f(x) = (-2(2sin(1\u03c0(-2|x + 3| - 3) - 1) - 1) - 3)/(2(2sin(1\u03c0(-2|x + 3| - 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"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1|(-3(4x^2 - 2)^2 + 5) + 2| + 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 total count of local extrema points for the function f(x) = 4(1(-3sin(2\u03c0x + 1) - 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": "Determine the total count of local extrema points for the function f(x) = 3((-1(1cos(1\u03c0x + 3) - 1))/(-1(1cos(1\u03c0x + 3) - 1) + 2))^2 - 2((-1(1cos(1\u03c0x + 3) - 1))/(-1(1cos(1\u03c0x + 3) - 1) + 2)) + 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": "How many local maxima does the function f(x) = 3|(3cos(3\u03c0(2|x + 2| + 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(-4(-1sin(1\u03c0x) - 1)^2 + 2(-1sin(1\u03c0x) - 1) + 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": "How many local maxima does the function f(x) = 2cos(2\u03c0(-1(3|x + 2| + 3)^2 + 2(3|x + 2| + 3) + 1)) 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": "Determine the count of local minima for the function f(x) = -2(-1sin(1\u03c0(-1x + 2) - 3) - 1)^2 + 5(-1sin(1\u03c0(-1x + 2) - 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": "Find the number of local maximum points for f(x) = 2sin(2\u03c0(-3|(-3|x + 1|)| - 1) - 3) + 2 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) = 2(1(-2cos(3\u03c0x) - 3) + 3)^2 - 2(1(-2cos(3\u03c0x) - 3) + 3) + 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 number of local maximum points for f(x) = -3(-2cos(2\u03c0(3x + 2) - 1) + 3)^2 - 5(-2cos(2\u03c0(3x + 2) - 1) + 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 maxima does the function f(x) = -1cos(3\u03c0(2|((-2x - 2)/(-1x + 3)) - 1| + 3)) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1sin(3\u03c0(-5(4x + 4) + 2)) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "399", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(1\u03c0(-1(-3x^2 + 1x - 5) + 3) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "289", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2cos(2\u03c0((3(-2|x - 3|) - 2)/(-2(-2|x - 3|) + 3)) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 5(-2sin(3\u03c0(3x^2 - 3x - 4) - 2) - 2)^2 + 1(-2sin(3\u03c0(3x^2 - 3x - 4) - 2) - 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "223", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -2|(3|(1x^2 - 5x)| - 3) - 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": "Find the number of local maximum points for f(x) = (-1(1(-1cos(2\u03c0x + 2) + 3)^2 + 3(-1cos(2\u03c0x + 2) + 3) + 4) - 1)/(3(1(-1cos(2\u03c0x + 2) + 3)^2 + 3(-1cos(2\u03c0x + 2) + 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) = 1(-1cos(2\u03c0(4x - 2) - 2) + 3)^2 - 1(-1cos(2\u03c0(4x - 2) - 2) + 3) - 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 minima does the function f(x) = -1(3|(2sin(1\u03c0x - 2) + 1) - 2| + 3)^2 + 5(3|(2sin(1\u03c0x - 2) + 1) - 2| + 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": "Find the total number of local minima and maxima for f(x) = (1((-1(-3cos(1\u03c0x - 2) + 3))/(1(-3cos(1\u03c0x - 2) + 3) + 1)))/(-2((-1(-3cos(1\u03c0x - 2) + 3))/(1(-3cos(1\u03c0x - 2) + 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 extrema (minima and maxima combined) does the function f(x) = (3((-2(-1sin(2\u03c0x - 2) - 2) + 1)/(1(-1sin(2\u03c0x - 2) - 2) + 1)) - 1)/(1((-2(-1sin(2\u03c0x - 2) - 2) + 1)/(1(-1sin(2\u03c0x - 2) - 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": "How many local minima does the function f(x) = 1sin(3\u03c0(5(-4x^2 - 5x - 2)^2 - 1(-4x^2 - 5x - 2) + 1) + 3) 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 minima does the function f(x) = 3sin(2\u03c0(-3(2x + 2)^2 + 1(2x + 2) + 4) - 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": "How many local maxima does the function f(x) = (1((1(-1x^2 + 2x + 3))/(3(-1x^2 + 2x + 3))) - 2)/(-1((1(-1x^2 + 2x + 3))/(3(-1x^2 + 2x + 3))) - 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": "Find the number of local maximum points for f(x) = (1((-2(-3sin(1\u03c0x - 3) + 2) - 3)/(1(-3sin(1\u03c0x - 3) + 2) + 2)) - 3)/(-1((-2(-3sin(1\u03c0x - 3) + 2) - 3)/(1(-3sin(1\u03c0x - 3) + 2) + 2)) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3(1sin(3\u03c0(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": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2((-1(1sin(2\u03c0x) - 2) + 2)/(-3(1sin(2\u03c0x) - 2) + 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": "Find the number of local maximum points for f(x) = (1(2sin(3\u03c0(-5x - 1) + 3)) + 1)/(2(2sin(3\u03c0(-5x - 1) + 3))) 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": "Find the number of local minimum points for f(x) = -5(-1sin(2\u03c0(-1|x - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1|(1cos(2\u03c0((-1x)/(2x - 3)) - 1) + 3) - 1| 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 total count of local extrema points for the function f(x) = 3(-3cos(1\u03c0(-1|x - 1| - 2) - 2) + 2)^2 - 1(-3cos(1\u03c0(-1|x - 1| - 2) - 2) + 2) - 5 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) = 4(3sin(1\u03c0((-2x - 2)/(-1x))) + 1) - 1 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": "Find the number of local maximum points for f(x) = 2sin(2\u03c0(-2((2x)/(3x - 2))^2 + 2((2x)/(3x - 2)) - 3) - 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 extrema (minima and maxima combined) does the function f(x) = (1(3cos(1\u03c0(-1x^2 - 2x + 4) - 1) - 2) - 1)/(1(3cos(1\u03c0(-1x^2 - 2x + 4) - 1) - 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 extrema (minima and maxima combined) does the function f(x) = -5(-2sin(3\u03c0(-2|x - 1|) + 2) - 2)^2 - 5(-2sin(3\u03c0(-2|x - 1|) + 2) - 2) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = (-1(-3(2cos(3\u03c0x - 3) - 3) - 1) - 3)/(-3(-3(2cos(3\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": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(3\u03c0(1|(2x^2 - 4x - 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": "235", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1|(1|(-2sin(3\u03c0x + 3) + 2) + 3|) + 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 maximum points for f(x) = -1(-2sin(1\u03c0(-2|x + 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": "Find the number of local minimum points for f(x) = -2sin(1\u03c0(-3(-2|x| - 1) - 5) + 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": "How many local maxima does the function f(x) = 1(-3cos(2\u03c0((1x + 1)/(3x)) - 1) - 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": "Find the total number of local minima and maxima for f(x) = (3(-5(1sin(1\u03c0x + 1) - 2) - 2))/(-2(-5(1sin(1\u03c0x + 1) - 2) - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "262", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (3(3|(-3sin(3\u03c0x + 3) + 2) - 3| + 2) - 1)/(3(3|(-3sin(3\u03c0x + 3) + 2) - 3| + 2) + 1) 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": "How many local extrema (minima and maxima combined) does the function f(x) = -2|(-3|(-5x^2 + 5x + 1) + 1| + 2)| - 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) = -3(-4(3cos(1\u03c0x - 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": "Find the number of local maximum points for f(x) = 3sin(3\u03c0(-4((3x + 1)/(2x + 2)) - 2) + 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 total count of local extrema points for the function f(x) = -1e^(-2(1|(2x + 3) - 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": "Find the number of local minimum points for f(x) = 1cos(1\u03c0((-3(-1x^2) - 2)/(1(-1x^2) - 2)) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-2(5(1cos(2\u03c0x + 2) - 2) - 3))/(-1(5(1cos(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": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3|((-1(-2cos(2\u03c0x - 3)) - 3)/(3(-2cos(2\u03c0x - 3)) + 3)) - 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": "How many local maxima does the function f(x) = -3cos(2\u03c0(-2|(-3x^2 - 1x - 1)| + 3) + 3) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "245", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3((-2(-2|x + 3| + 3) + 2)/(2(-2|x + 3| + 3) - 1))^2 + 3((-2(-2|x + 3| + 3) + 2)/(2(-2|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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(1sin(3\u03c0(-2|x|) - 3) - 3) - 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 maximum points for f(x) = -2(-2cos(1\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": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(4(-1cos(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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3sin(2\u03c0((-3(-2|x - 1| + 1) - 1)/(1(-2|x - 1| + 1) + 1)) - 1) - 1 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": "How many local minima does the function f(x) = (-3((3(3sin(1\u03c0x - 2) + 3) - 2)/(2(3sin(1\u03c0x - 2) + 3) + 2)) + 1)/(-2((3(3sin(1\u03c0x - 2) + 3) - 2)/(2(3sin(1\u03c0x - 2) + 3) + 2)) + 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) = -3e^(-1(5(-1|x + 2| + 3)^2 + 4(-1|x + 2| + 3) - 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3cos(1\u03c0((2((1x - 2)/(-1x - 1)) - 3)/(-1((1x - 2)/(-1x - 1)) - 1)) + 3) + 1 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 count of local maxima for the function f(x) = -2|(-3e^(1(5x + 2)) - 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": "Determine the count of local maxima for the function f(x) = (1(3sin(3\u03c0(-4x^2 + 1x - 1) - 3) + 1) + 2)/(-3(3sin(3\u03c0(-4x^2 + 1x - 1) - 3) + 1) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "258", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 5(-2cos(3\u03c0(-2|x - 2| - 3)) + 1)^2 - 1(-2cos(3\u03c0(-2|x - 2| - 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": "Find the number of local minimum points for f(x) = 2cos(1\u03c0(-1|((-1x + 1)/(2x - 1)) + 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 minima for the function f(x) = 2cos(3\u03c0(2(-1x^2 - 3)^2 + 4(-1x^2 - 3) - 5) - 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "163", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1sin(1\u03c0((2(-3|x - 2| + 1) + 2)/(3(-3|x - 2| + 1) - 2)) + 1) - 2 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) = (1(-3sin(3\u03c0(-1|x - 3|) - 2) + 3) + 1)/(-3(-3sin(3\u03c0(-1|x - 3|) - 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 number of local minimum points for f(x) = (-2(-1sin(3\u03c0(5x^2 + 2x - 4) - 2) - 3) + 3)/(-2(-1sin(3\u03c0(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": "321", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 4(2cos(3\u03c0((-2x)/(2x + 3)) + 2) - 2)^2 - 2(2cos(3\u03c0((-2x)/(2x + 3)) + 2) - 2) + 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": "How many local minima does the function f(x) = 2|(1cos(2\u03c0(1x^2 + 4x - 5) - 2)) + 3| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "211", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1(-1(1sin(2\u03c0x + 3) - 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 maxima for the function f(x) = (3(-5(2e^(-2x) + 3)^2 - 3(2e^(-2x) + 3) + 2) - 2)/(-3(-5(2e^(-2x) + 3)^2 - 3(2e^(-2x) + 3) + 2) - 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(1\u03c0(4(3x - 5) + 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": "How many local minima does the function f(x) = (-3(-2cos(1\u03c0(2x - 5) + 2) - 3) + 1)/(-2(-2cos(1\u03c0(2x - 5) + 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": "Determine the total count of local extrema points for the function f(x) = -2cos(1\u03c0(3(-3x^2 + 2x + 2)^2 + 4(-3x^2 + 2x + 2)) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "237", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3(2|(2x^2 + 2x - 5) + 2|)^2 - 3(2|(2x^2 + 2x - 5) + 2|) - 5 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 maxima for the function f(x) = 1(5(-2cos(3\u03c0x + 1) + 1)^2 - 1(-2cos(3\u03c0x + 1) + 1) - 2) + 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": "Find the number of local maximum points for f(x) = 1cos(3\u03c0(2|(3x + 1)| - 2) - 2) - 3 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": "Determine the count of local minima for the function f(x) = 2cos(3\u03c0(3|((-3x + 2)/(3x - 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": "Find the number of local maximum points for f(x) = (-1(3sin(3\u03c0((-2x - 1)/(1x - 2)) - 1) + 3) - 1)/(-1(3sin(3\u03c0((-2x - 1)/(1x - 2)) - 1) + 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (1(-3|(-3cos(1\u03c0x + 3) + 1) - 1| + 2) - 1)/(-3(-3|(-3cos(1\u03c0x + 3) + 1) - 1| + 2) + 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": "How many local minima does the function f(x) = -1cos(2\u03c0(-1(-1x + 1)^2 - 4(-1x + 1)) - 3) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "218", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2(-2sin(1\u03c0(-1x + 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": "Determine the count of local maxima for the function f(x) = 5(-2cos(3\u03c0((-2x - 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": "16", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -4(3cos(1\u03c0(-3|x - 1| - 2) + 2) + 1)^2 - 3(3cos(1\u03c0(-3|x - 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": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (-1(-2cos(2\u03c0((2x)/(-2x + 3)))) - 1)/(2(-2cos(2\u03c0((2x)/(-2x + 3)))) - 1) 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 minima does the function f(x) = 4(2cos(1\u03c0((-2x)/(3x + 2)) + 1) + 2)^2 - 3(2cos(1\u03c0((-2x)/(3x + 2)) + 1) + 2) - 5 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) = (-1(-3sin(3\u03c0(2|x| - 2)) + 3) + 2)/(3(-3sin(3\u03c0(2|x| - 2)) + 3) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(2(-3sin(1\u03c0x + 1) - 2) - 4)^2 - 5(2(-3sin(1\u03c0x + 1) - 2) - 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 extrema (minima and maxima combined) does the function f(x) = 3sin(1\u03c0(2|(-3x^2 + 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": "453", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 3sin(3\u03c0(1(-1x^2 - 1x - 2)^2 + 5(-1x^2 - 1x - 2) - 4) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "162", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3cos(1\u03c0(-2(3|x - 3| - 2) + 4) - 3) + 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": "Find the number of local minimum points for f(x) = 3|(-3sin(1\u03c0((2x - 2)/(2x - 1)) - 2) - 3) - 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 total number of local minima and maxima for f(x) = -2e^(-1(3|(5x^2 + 5x - 1) - 1| - 3)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2(-5(-2sin(3\u03c0x + 1))^2 - 4(-2sin(3\u03c0x + 1)) - 2) - 4 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 extrema (minima and maxima combined) does the function f(x) = (-3((3(1cos(2\u03c0x) - 1) - 2)/(-2(1cos(2\u03c0x) - 1) + 3)) - 1)/(-3((3(1cos(2\u03c0x) - 1) - 2)/(-2(1cos(2\u03c0x) - 1) + 3)) + 2) 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 maxima does the function f(x) = 1cos(1\u03c0((-2(-3|x + 1| + 1))/(-2(-3|x + 1| + 1) + 1)) - 2) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = (-1(2e^(-1(2x^2 - 1x + 1)) - 3) + 1)/(-3(2e^(-1(2x^2 - 1x + 1)) - 3) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(2(4x^2 + 2x + 5) - 4) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "263", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2|(1sin(1\u03c0(1|x| + 3) - 3) - 3) - 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) = 3(-1cos(3\u03c0(4x^2 + 1x + 5))) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (2(3sin(3\u03c0((-2x + 3)/(-3x + 1)) - 2) - 1) + 3)/(1(3sin(3\u03c0((-2x + 3)/(-3x + 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": "Find the total number of local minima and maxima for f(x) = -1(-1cos(2\u03c0(-3x)) - 2)^2 - 4(-1cos(2\u03c0(-3x)) - 2) + 4 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": "Determine the count of local maxima for the function f(x) = 1|(-3(2cos(1\u03c0x + 3) + 1)^2 - 1(2cos(1\u03c0x + 3) + 1) + 4) - 1| + 3 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 extrema (minima and maxima combined) does the function f(x) = 2|(-4(-1cos(3\u03c0x - 3) - 3)^2 + 4(-1cos(3\u03c0x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(1(-2sin(2\u03c0x - 1) - 2)^2 + 5(-2sin(2\u03c0x - 1) - 2) - 3)^2 + 3(1(-2sin(2\u03c0x - 1) - 2)^2 + 5(-2sin(2\u03c0x - 1) - 2) - 3) + 4 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) = 2|(-5(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) + 5) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1((-2(3sin(2\u03c0x) + 3) + 3)/(2(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": "120", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (3((2(-2sin(2\u03c0x) - 2))/(3(-2sin(2\u03c0x) - 2) - 1)) - 2)/(2((2(-2sin(2\u03c0x) - 2))/(3(-2sin(2\u03c0x) - 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": "Find the total number of local minima and maxima for f(x) = (-2(3(3cos(3\u03c0x - 3))^2 + 5) - 1)/(-1(3(3cos(3\u03c0x - 3))^2 + 5) + 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": "Determine the count of local minima for the function f(x) = 5(3sin(2\u03c0((1x - 1)/(-1x - 1)) - 3) - 3)^2 + 1(3sin(2\u03c0((1x - 1)/(-1x - 1)) - 3) - 3) - 1 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 total number of local minima and maxima for f(x) = -3|(5(-2sin(1\u03c0x - 1) - 1) - 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": "How many local maxima does the function f(x) = 5(-3cos(1\u03c0(-1x - 1) + 3) + 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": "Determine the count of local maxima for the function f(x) = 5(-3cos(3\u03c0(2x^2 + 3x - 4) + 2) + 2)^2 - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 4(3(1sin(3\u03c0x + 2) - 1)^2 + 1(1sin(3\u03c0x + 2) - 1) + 4) + 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": "Find the number of local maximum points for f(x) = -3sin(1\u03c0(-3(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3) - 3) + 2 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) = 5(1(-2x^2 - 1)^2 + 4(-2x^2 - 1))^2 - 4(1(-2x^2 - 1)^2 + 4(-2x^2 - 1)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2|(-1cos(2\u03c0((-3x - 1)/(3x + 3)) + 2) + 1) - 3| - 3 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": "How many local maxima does the function f(x) = 1(5(1cos(1\u03c0x - 3) - 3) + 1)^2 - 4(5(1cos(1\u03c0x - 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": "Determine the total count of local extrema points for the function f(x) = (2(-2cos(2\u03c0(1x^2 + 1x + 3) + 1) - 2) - 2)/(2(-2cos(2\u03c0(1x^2 + 1x + 3) + 1) - 2) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "403", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3(-2cos(3\u03c0(4x + 2) + 1) - 2) - 5 have 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) = 5(-4(1cos(1\u03c0x + 3) - 3) + 3)^2 + 5(-4(1cos(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": "Determine the count of local maxima for the function f(x) = (-1(-2cos(1\u03c0(-5x + 4) + 3) + 3))/(3(-2cos(1\u03c0(-5x + 4) + 3) + 3) - 2) 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) = -1(1cos(3\u03c0(3x - 3) - 1) + 2)^2 - 5(1cos(3\u03c0(3x - 3) - 1) + 2) - 5 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": "How many local minima does the function f(x) = 1(-2|(2sin(2\u03c0x - 2) - 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": "How many local maxima does the function f(x) = 3cos(2\u03c0(2|(-3|x - 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2sin(1\u03c0(4(-3|x - 3|)) - 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": "Determine the count of local minima for the function f(x) = -1cos(2\u03c0(-2(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) - 5) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "245", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -2(-3cos(3\u03c0(1|x - 1|)))^2 + 4(-3cos(3\u03c0(1|x - 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 minima does the function f(x) = 1|(2cos(1\u03c0(-2x^2 + 4x - 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": "202", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (-2(-1sin(3\u03c0((1x - 2)/(-1x + 3)) - 3) + 2) - 3)/(-3(-1sin(3\u03c0((1x - 2)/(-1x + 3)) - 3) + 2) + 2) 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) = -5(3|(2cos(1\u03c0x - 2) + 2) - 3| - 3)^2 - 5(3|(2cos(1\u03c0x - 2) + 2) - 3| - 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) = -2|(-3cos(1\u03c0(-2|x - 2| + 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": "81", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4(-5(1sin(3\u03c0x - 3) + 2) + 1)^2 + 1(-5(1sin(3\u03c0x - 3) + 2) + 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": "How many local minima does the function f(x) = -1cos(1\u03c0(1(-1|x + 1| - 2) - 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 number of local minimum points for f(x) = -3sin(2\u03c0(3|(2|x| - 2) + 2| + 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": "Determine the total count of local extrema points for the function f(x) = 2(1sin(1\u03c0(2x^2) - 3) - 1) + 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) = 3|(3sin(1\u03c0(-3|x - 3| - 1) - 2) + 2) - 1| - 3 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) = 1|(-1cos(1\u03c0(-5x^2 - 2) + 1)) + 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "252", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-2(1sin(1\u03c0(4x - 1) + 2) + 3) - 2)/(2(1sin(1\u03c0(4x - 1) + 2) + 3) - 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) = 3(-2|(-1cos(3\u03c0x - 2) + 2) - 2| + 3)^2 - 4(-2|(-1cos(3\u03c0x - 2) + 2) - 2| + 3) - 4 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) = -2(-2(1sin(1\u03c0x + 2) - 3)) - 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 maxima does the function f(x) = -3|(-1cos(1\u03c0(1x - 1) + 2) + 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": "Determine the total count of local extrema points for the function f(x) = 2e^(-2(3|(1x^2 + 3x)| + 3)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 4(-2(-2cos(3\u03c0x - 1) + 1) - 2)^2 + 1(-2(-2cos(3\u03c0x - 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 2cos(2\u03c0(5(5x^2 - 3x) + 3) - 1) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "478", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2(-1cos(1\u03c0(-1x^2 - 2x - 2) + 3) - 3)^2 - 4(-1cos(1\u03c0(-1x^2 - 2x - 2) + 3) - 3) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(-1sin(1\u03c0((-1x - 2)/(-2x + 3)) + 2) - 1)^2 + 3(-1sin(1\u03c0((-1x - 2)/(-2x + 3)) + 2) - 1) + 4 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 minima does the function f(x) = 1cos(2\u03c0(-5(1x - 2) + 2) - 2) + 2 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 maxima does the function f(x) = -1sin(1\u03c0(2|(3|x - 2| - 2) + 2| - 3) - 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) = 1cos(3\u03c0(-4((-2x + 1)/(2x + 3))^2 - 2((-2x + 1)/(2x + 3)) + 1) + 2) - 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 total count of local extrema points for the function f(x) = 3sin(2\u03c0((-1(1x^2 + 4x) + 2)/(-2(1x^2 + 4x) + 1)) - 2) + 2 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 extrema (minima and maxima combined) does the function f(x) = -2sin(3\u03c0(-2|(-3x^2 + 5x + 2) - 2| + 1) + 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "497", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2((3((2x + 2)/(3x + 3)) - 2)/(3((2x + 2)/(3x + 3)) - 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": "How many local maxima does the function f(x) = (-1(-2sin(1\u03c0(-4x - 5)) - 3) + 2)/(3(-2sin(1\u03c0(-4x - 5)) - 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": "Find the total number of local minima and maxima for f(x) = 1e^(1(2|(5x^2 + 5x + 1) - 3| - 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(1\u03c0(5((3x + 3)/(-2x - 3)) - 4) - 1) 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 total count of local extrema points for the function f(x) = 1((-1(-2|x + 1| + 2) + 3)/(-2(-2|x + 1| + 2) - 1))^2 + 4((-1(-2|x + 1| + 2) + 3)/(-2(-2|x + 1| + 2) - 1)) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1cos(2\u03c0(4((2x - 1)/(-3x - 3)) + 5)) - 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) = 2|(-1sin(1\u03c0(-3x - 5)) - 2) + 3| - 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) = 2|(3|(-1sin(1\u03c0x + 3) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3sin(2\u03c0((1(3x + 2) - 1)/(1(3x + 2) + 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": "Find the total number of local minima and maxima for f(x) = 1cos(3\u03c0(-3|(-1x^2 + 2x - 2) - 1| + 2)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "515", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 4(-1cos(3\u03c0((1x - 2)/(-3x + 4)) - 1) + 1) + 3 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) = -1(3cos(3\u03c0(3|x - 1| - 2) + 3) - 1)^2 + 4(3cos(3\u03c0(3|x - 1| - 2) + 3) - 1) + 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) = -1(3sin(3\u03c0(2x^2 - 1) - 1) - 2)^2 + 5(3sin(3\u03c0(2x^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": "186", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(3sin(1\u03c0(4x - 5) - 2) + 2) - 1)/(-3(3sin(1\u03c0(4x - 5) - 2) + 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": "Determine the total count of local extrema points for the function f(x) = 3|(3|(2sin(3\u03c0x - 2) + 1) + 1|) - 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": "Find the number of local maximum points for f(x) = -2cos(1\u03c0((3(-2|x - 1| + 3) + 3)/(-1(-2|x - 1| + 3) - 3)) + 3) + 1 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": "How many local extrema (minima and maxima combined) does the function f(x) = (3((3(-3x + 2) - 1)/(2(-3x + 2) + 3)) + 3)/(-2((3(-3x + 2) - 1)/(2(-3x + 2) + 3)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "407", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|(5(-2cos(3\u03c0x + 2) - 3) + 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": "How many local maxima does the function f(x) = -3|(-2cos(1\u03c0(-3x - 4) - 3) - 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 count of local maxima for the function f(x) = 2(-1(-1sin(2\u03c0x + 3) - 3)^2 + 3(-1sin(2\u03c0x + 3) - 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": "Find the total number of local minima and maxima for f(x) = -1(2sin(1\u03c0((-3x + 2)/(-1x + 2)) - 2) + 1) - 4 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) = -3(3sin(3\u03c0(-3x - 4) - 3) - 2)^2 - 4(3sin(3\u03c0(-3x - 4) - 3) - 2) - 3 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": "Determine the count of local minima for the function f(x) = 1sin(1\u03c0(-3((2x - 3)/(-3x - 1))^2 + 4((2x - 3)/(-3x - 1)) - 1) + 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": "Find the number of local minimum points for f(x) = 1((2(2sin(1\u03c0x + 1) - 1) - 3)/(2(2sin(1\u03c0x + 1) - 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3(-2|(3sin(1\u03c0x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -4(2sin(3\u03c0(-1|x + 2|) + 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": "Determine the total count of local extrema points for the function f(x) = 3|(5(-2cos(3\u03c0x - 3))) - 1| - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "113", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1sin(3\u03c0(-3|(-3x + 2)|) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "541", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3(-2(-1cos(3\u03c0x - 3) + 3) - 5) + 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": "How many local maxima does the function f(x) = 2cos(3\u03c0(-1(-1x)^2 - 4(-1x) - 3) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "270", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = -1(-1cos(2\u03c0(-1|x|)) - 2)^2 - 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 extrema (minima and maxima combined) does the function f(x) = (-3(1sin(1\u03c0(2x + 3) - 2) - 2) + 1)/(-2(1sin(1\u03c0(2x + 3) - 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": "How many local minima does the function f(x) = -2|(1sin(1\u03c0((-2x + 1)/(-1x + 3)) - 1) + 3)| + 1 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 maxima does the function f(x) = (2((-1(-2cos(2\u03c0x + 2) - 3) - 2)/(-3(-2cos(2\u03c0x + 2) - 3))) - 1)/(1((-1(-2cos(2\u03c0x + 2) - 3) - 2)/(-3(-2cos(2\u03c0x + 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": "Determine the total count of local extrema points for the function f(x) = -1sin(1\u03c0((-2((-1x + 1)/(-3x + 1)) + 3)/(2((-1x + 1)/(-3x + 1)) + 2)) - 3) + 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 count of local maxima for the function f(x) = -5(3sin(2\u03c0(5x + 4) - 3)) 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 minima does the function f(x) = (-2(1cos(2\u03c0(1x - 3)) + 1) - 2)/(3(1cos(2\u03c0(1x - 3)) + 1) + 3) 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) = 1((-1(2|x - 3| - 1) - 3)/(-2(2|x - 3| - 1) + 3))^2 - 5((-1(2|x - 3| - 1) - 3)/(-2(2|x - 3| - 1) + 3)) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2|(-1e^(2(1x - 4)))| + 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) = -1cos(1\u03c0((3(3x^2 + 4x + 1) + 2)/(2(3x^2 + 4x + 1) - 3)) - 1) - 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 maximum points for f(x) = -5(5(-3|x - 3| + 2)^2 + 1(-3|x - 3| + 2) - 4)^2 - 4(5(-3|x - 3| + 2)^2 + 1(-3|x - 3| + 2) - 4) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2|(-2e^(-2(-3x + 2)) - 1)| 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 minima for the function f(x) = -5(-1(-2cos(2\u03c0x - 3)) - 2) + 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": "Determine the count of local maxima for the function f(x) = -1|(-5(3sin(3\u03c0x + 3) + 3) - 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": "How many local minima does the function f(x) = -2sin(1\u03c0(2(5x) + 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": "Determine the total count of local extrema points for the function f(x) = -2|((-1(2sin(3\u03c0x + 3) + 2) - 3)/(1(2sin(3\u03c0x + 3) + 2) - 2)) + 1| - 1 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 total count of local extrema points for the function f(x) = -1((2(2sin(3\u03c0x - 1) - 3) - 2)/(-2(2sin(3\u03c0x - 1) - 3) + 3))^2 + 3((2(2sin(3\u03c0x - 1) - 3) - 2)/(-2(2sin(3\u03c0x - 1) - 3) + 3)) + 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) = 3((-1(1sin(2\u03c0x - 3) + 2) - 3)/(-2(1sin(2\u03c0x - 3) + 2) - 2)) + 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) = 2sin(3\u03c0(3|(-5x^2 + 4x) - 3| - 2) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "245", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2(-1cos(1\u03c0((-3x - 1)/(2x + 1)) - 3) - 3)^2 - 2(-1cos(1\u03c0((-3x - 1)/(2x + 1)) - 3) - 3) - 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 extrema (minima and maxima combined) does the function f(x) = -3(3sin(1\u03c0((1x)/(-3x - 2)) + 2))^2 - 3(3sin(1\u03c0((1x)/(-3x - 2)) + 2)) - 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": "Find the total number of local minima and maxima for f(x) = (1(-2|(2cos(2\u03c0x + 3) - 3) - 1| - 2) + 3)/(-3(-2|(2cos(2\u03c0x + 3) - 3) - 1| - 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": "Find the number of local minimum points for f(x) = 5(-1(-1x^2 - 2x + 4)^2 - 4(-1x^2 - 2x + 4) - 3)^2 - 5(-1(-1x^2 - 2x + 4)^2 - 4(-1x^2 - 2x + 4) - 3) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(1cos(1\u03c0(4x - 4)) - 1) - 3| + 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 total count of local extrema points for the function f(x) = -3(1sin(2\u03c0((-1x + 1)/(-1x + 2)) - 3) + 1) - 4 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 extrema (minima and maxima combined) does the function f(x) = 1|(1(-2sin(2\u03c0x + 2) + 2)^2 + 2(-2sin(2\u03c0x + 2) + 2) - 2) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3|(3sin(1\u03c0((-2x - 3)/(1x + 3))) + 1) - 2| + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(2\u03c0(-4(-1|x - 1| + 3)^2 + 3(-1|x - 1| + 3) + 1) - 2) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "459", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 4(4(-3cos(2\u03c0x - 1)) + 4)^2 - 1(4(-3cos(2\u03c0x - 1)) + 4) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 4(2sin(1\u03c0(3x^2 - 3x - 3) + 1)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "565", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3(-3cos(1\u03c0(-2|x + 1|) - 1) - 3)^2 + 4(-3cos(1\u03c0(-2|x + 1|) - 1) - 3) - 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 total number of local minima and maxima for f(x) = 3(-2sin(1\u03c0((-2x - 3)/(-1x)) - 3) - 2)^2 - 2(-2sin(1\u03c0((-2x - 3)/(-1x)) - 3) - 2) + 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": "Find the number of local maximum points for f(x) = -1sin(1\u03c0(2|((-2x - 2)/(1x + 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": "Find the total number of local minima and maxima for f(x) = 2cos(2\u03c0((2(-1|x| + 2) + 3)/(2(-1|x| + 2) - 1))) + 3 where -10 \u2264 x \u2264 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) = 1(3cos(2\u03c0(3x + 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3e^(2(-3|((2x - 2)/(-1x - 3))| - 3)) - 1 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) = -2sin(2\u03c0(-4(-2x^2 + 3x + 3) + 3) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "249", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 1cos(1\u03c0(1|((2x + 1)/(3x - 1)) - 3| + 3) + 2) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3sin(3\u03c0(-1(1x^2 - 2x + 5) - 4) - 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "559", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 2(2(3sin(1\u03c0x - 2) - 3)^2 - 4(3sin(1\u03c0x - 2) - 3) + 4) + 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": "Find the number of local minimum points for f(x) = 4(-2|(4x^2 + 2x - 3) + 2| + 2)^2 + 4(-2|(4x^2 + 2x - 3) + 2| + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 1(-3(2sin(2\u03c0x - 2) + 3) - 2)^2 - 3(-3(2sin(2\u03c0x - 2) + 3) - 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 minima for the function f(x) = 2(-1|(1cos(2\u03c0x + 3))| + 2)^2 - 5(-1|(1cos(2\u03c0x + 3))| + 2) + 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) = (-2(3sin(1\u03c0(5x^2 + 3x - 1) - 1) + 3) + 2)/(2(3sin(1\u03c0(5x^2 + 3x - 1) - 1) + 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "501", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1sin(2\u03c0(1|(4x + 2) + 3| - 1) - 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": "Find the total number of local minima and maxima for f(x) = 5((2(3cos(3\u03c0x) + 1) + 3)/(-2(3cos(3\u03c0x) + 1)))^2 - 5((2(3cos(3\u03c0x) + 1) + 3)/(-2(3cos(3\u03c0x) + 1))) - 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": "Determine the count of local maxima for the function f(x) = 2cos(1\u03c0(-1|(4x^2 - 3x) - 3| + 3)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "287", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1(2sin(3\u03c0(-2x^2 - 2)))^2 - 3(2sin(3\u03c0(-2x^2 - 2))) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(1\u03c0((-1(2|x - 3| - 2) - 2)/(-3(2|x - 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": "14", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-3(-4(2sin(2\u03c0x) + 1) + 2))/(-2(-4(2sin(2\u03c0x) + 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": "Determine the total count of local extrema points for the function f(x) = 3|(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": "21", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -3(2|(-3sin(3\u03c0x + 1) - 2) - 1| - 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) = -3(-3(2sin(3\u03c0x - 1) + 3) + 2)^2 + 3(-3(2sin(3\u03c0x - 1) + 3) + 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 total count of local extrema points for the function f(x) = -1|(-1sin(3\u03c0(-3|x + 2| + 3) + 3) - 1) - 3| - 3 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) = 4(-2(3cos(3\u03c0x + 2) - 3) + 1)^2 + 3(-2(3cos(3\u03c0x + 2) - 3) + 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) = 4(3cos(2\u03c0(-4x^2 - 3x - 1) - 3) - 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "259", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4(2((-3x - 3)/(-2x - 2))^2 + 1((-3x - 3)/(-2x - 2)) - 2)^2 + 2(2((-3x - 3)/(-2x - 2))^2 + 1((-3x - 3)/(-2x - 2)) - 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 count of local maxima for the function f(x) = 1|(3sin(1\u03c0((1x)/(-3x)) - 3) - 3) - 1| - 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": "How many local maxima does the function f(x) = 2cos(1\u03c0(-4(-2x - 3) + 4) - 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": "Determine the count of local minima for the function f(x) = -1|(1|(2sin(3\u03c0x + 2) - 1) + 3| + 1) - 2| + 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 count of local maxima for the function f(x) = -3(2e^(-1(3|x + 1| + 2))) - 5 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 maximum points for f(x) = -3sin(1\u03c0(4(-4x^2 - 1x + 4)^2 - 3(-4x^2 - 1x + 4) + 4) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4((-2(2sin(1\u03c0x - 3) + 2) + 1)/(-3(2sin(1\u03c0x - 3) + 2) - 3))^2 + 4((-2(2sin(1\u03c0x - 3) + 2) + 1)/(-3(2sin(1\u03c0x - 3) + 2) - 3)) - 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 total number of local minima and maxima for f(x) = 2cos(3\u03c0(1|((2x + 2)/(-3x - 1))| - 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": "How many local minima does the function f(x) = -1sin(3\u03c0(4((-3x - 3)/(2x + 3)) - 1) - 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 minima does the function f(x) = (-1(1sin(1\u03c0(-3|x + 2| - 1)) - 1) + 3)/(-1(1sin(1\u03c0(-3|x + 2| - 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": "Determine the count of local minima for the function f(x) = 4(3(3cos(2\u03c0x + 3) - 2) + 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 total number of local minima and maxima for f(x) = (-2((-2(3sin(3\u03c0x - 1) + 2))/(3(3sin(3\u03c0x - 1) + 2) + 3)))/(1((-2(3sin(3\u03c0x - 1) + 2))/(3(3sin(3\u03c0x - 1) + 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": "Find the number of local maximum points for f(x) = (-1(-1sin(3\u03c0(2x^2 + 3x - 4)) - 3))/(3(-1sin(3\u03c0(2x^2 + 3x - 4)) - 3) + 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": "Determine the total count of local extrema points for the function f(x) = -2cos(2\u03c0(-1(3x^2 + 3x - 1) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "455", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3sin(2\u03c0(5(2x^2 + 5x + 2)^2 - 2(2x^2 + 5x + 2) - 5)) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "193", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -4(1sin(3\u03c0(2|x - 3|) - 1) + 3) + 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": "Determine the count of local minima for the function f(x) = 2(3cos(3\u03c0(-4x^2 - 1x) + 1)) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "237", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (-1(-5(3sin(3\u03c0x) - 2)) + 1)/(1(-5(3sin(3\u03c0x) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1|((2(2cos(2\u03c0x + 2) - 2) - 3)/(-1(2cos(2\u03c0x + 2) - 2) + 2)) - 1| + 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 minima does the function f(x) = -1|(5(4x^2 + 5x - 3)^2 + 3(4x^2 + 5x - 3) - 4) + 3| - 2 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) = 2cos(3\u03c0(-5((1x - 3)/(-2x - 2)) + 3) - 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": "Find the total number of local minima and maxima for f(x) = -3|(1|(-2cos(1\u03c0x + 1) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -3cos(1\u03c0(4(-2x + 1)^2 - 5(-2x + 1) + 4) - 2) have in the interval [-10, 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) = 4(-1sin(1\u03c0((-1x - 2)/(1x + 1))) + 1)^2 - 3(-1sin(1\u03c0((-1x - 2)/(1x + 1))) + 1) - 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 minima does the function f(x) = 1|(2(3sin(2\u03c0x + 1) + 1)^2 + 2(3sin(2\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = (1(-3sin(1\u03c0(-2x^2 - 1x - 4) + 1) + 3))/(-1(-3sin(1\u03c0(-2x^2 - 1x - 4) + 1) + 3) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "134", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2cos(1\u03c0(-5(4x^2 + 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": "496", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 5(1(-2cos(3\u03c0x) + 1) - 3) + 5 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) = 2cos(1\u03c0(4(-5x^2 - 3x)^2 + 4(-5x^2 - 3x) + 4) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = (-1(3sin(2\u03c0((-2x - 1)/(2x - 1)) + 3) - 1) + 3)/(1(3sin(2\u03c0((-2x - 1)/(2x - 1)) + 3) - 1) + 1) 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) = 1cos(1\u03c0(-3|(3|x| - 3) + 1| - 3) + 3) + 3 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 count of local maxima for the function f(x) = -1(4(-2cos(1\u03c0x - 3) - 2)^2 + 2(-2cos(1\u03c0x - 3) - 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": "Determine the count of local maxima for the function f(x) = 5(-2cos(3\u03c0(3x^2 + 4x + 3) - 2) - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "261", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (-1(-1|(1cos(2\u03c0x + 1) - 3) + 2| + 3) - 2)/(2(-1|(1cos(2\u03c0x + 1) - 3) + 2| + 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": "How many local maxima does the function f(x) = 3|(2(-1cos(3\u03c0x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (2(-3sin(3\u03c0(-1|x| + 2) + 1) - 1) - 2)/(1(-3sin(3\u03c0(-1|x| + 2) + 1) - 1)) 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|(1(-3sin(2\u03c0x + 3) - 3)^2 - 2(-3sin(2\u03c0x + 3) - 3) - 5) + 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) = 1|(-1sin(1\u03c0(2|x - 2|) + 1) + 1) + 3| - 3 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 extrema (minima and maxima combined) does the function f(x) = 5(3|(2cos(3\u03c0x - 1) - 2) + 3| + 1)^2 - 3(3|(2cos(3\u03c0x - 1) - 2) + 3| + 1) + 1 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 minima does the function f(x) = -5(-3cos(2\u03c0((-2x - 2)/(-1x + 3)) + 1) - 1) - 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) = -3|(2sin(1\u03c0(5x^2 + 5x - 4) - 1)) + 1| + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "477", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(2\u03c0(-4(-1x - 1)^2 - 2(-1x - 1) - 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "257", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 5(3|(-3cos(3\u03c0x + 2) - 3) - 2| + 1) + 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 extrema (minima and maxima combined) does the function f(x) = 1(4(3|x - 2|)^2 - 2(3|x - 2|) + 1)^2 - 4(4(3|x - 2|)^2 - 2(3|x - 2|) + 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1cos(3\u03c0(2(3|x| + 3) + 1) + 2) - 2 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 minimum points for f(x) = -2(-3sin(2\u03c0(-1|x - 2| - 3) + 1) - 2)^2 - 3(-3sin(2\u03c0(-1|x - 2| - 3) + 1) - 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": "Find the number of local minimum points for f(x) = -5(1|(2sin(1\u03c0x + 1) + 1) + 2| + 1)^2 + 5(1|(2sin(1\u03c0x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1|((-1(-2cos(2\u03c0x - 3)))/(1(-2cos(2\u03c0x - 3)) + 2)) - 1| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2sin(2\u03c0(-2|(1x^2 - 1x - 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": "573", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = 1cos(2\u03c0((-3(3x) - 2)/(-2(3x) - 1)) - 2) + 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": "How many local minima does the function f(x) = (-1(1sin(3\u03c0(4x^2 + 4x + 5) + 3) - 3) + 1)/(2(1sin(3\u03c0(4x^2 + 4x + 5) + 3) - 3) + 1) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "169", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2|(1cos(2\u03c0(-5x - 4) - 1) + 2) + 1| - 1 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 minimum points for f(x) = -5(-1|(-1sin(3\u03c0x + 3) + 3) + 1| + 2)^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": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(3\u03c0(-3((1x - 1)/(3x - 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": "Find the total number of local minima and maxima for f(x) = 1|(-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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2(-1(1cos(1\u03c0x) - 2)^2 - 3(1cos(1\u03c0x) - 2) - 5) - 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": "Determine the count of local maxima for the function f(x) = 2cos(2\u03c0(-3(-3x + 2) - 2) - 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 maxima for the function f(x) = -1sin(2\u03c0(-5(2x^2 + 5x + 4)^2 - 5(2x^2 + 5x + 4) + 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": "Find the number of local minimum points for f(x) = -3|((1(3|x|) - 1)/(2(3|x|) - 1)) - 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2sin(1\u03c0((-2(1x - 4) - 2)/(1(1x - 4))) - 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 extrema (minima and maxima combined) does the function f(x) = 2e^(2(-3|(2|x - 3|)|)) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-1(-3|(2cos(2\u03c0x + 1)) + 2| + 2) - 3)/(2(-3|(2cos(2\u03c0x + 1)) + 2| + 2) - 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) = 2(3|(-1sin(3\u03c0x - 3) + 2) - 3| - 1)^2 + 5(3|(-1sin(3\u03c0x - 3) + 2) - 3| - 1) - 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 maxima for the function f(x) = -3|(1(-3cos(2\u03c0x + 2) + 3)^2 + 3(-3cos(2\u03c0x + 2) + 3) + 4) - 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 minima for the function f(x) = -4(-3sin(3\u03c0(1x^2 + 4x + 3) + 3) - 1)^2 - 2(-3sin(3\u03c0(1x^2 + 4x + 3) + 3) - 1) - 1 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) = 2cos(1\u03c0(-3(-3x - 3) + 5) - 1) + 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": "Find the total number of local minima and maxima for f(x) = 1sin(3\u03c0(-2(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": "120", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1sin(1\u03c0((3(-4x^2 + 2x - 1))/(-1(-4x^2 + 2x - 1))) + 3) + 3 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(-1e^(-2(2x - 3)) + 1) + 3)/(3(-1e^(-2(2x - 3)) + 1) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (-3(-3(-1cos(3\u03c0x + 3) - 2)) - 3)/(-1(-3(-1cos(3\u03c0x + 3) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -1|(2|(-2cos(3\u03c0x - 2) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(3|(-3x^2 + 2) + 3|) - 1| + 2 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) = -2(-1sin(1\u03c0(-3x^2 - 3x - 3) - 3) - 1)^2 + 4(-1sin(1\u03c0(-3x^2 - 3x - 3) - 3) - 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "567", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(5(-2sin(3\u03c0x - 3) - 3) + 4) - 3)/(3(5(-2sin(3\u03c0x - 3) - 3) + 4) - 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) = -1(-5(-1sin(2\u03c0x - 2) + 2) + 4) + 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 minimum points for f(x) = 4(2cos(2\u03c0(-1|x - 2| - 2) + 3) - 1) - 4 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) = -1sin(1\u03c0(-2|(2x^2 - 2x + 2)| - 3) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "284", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -4(2sin(3\u03c0(1|x - 1| - 1) - 3) + 3)^2 - 2(2sin(3\u03c0(1|x - 1| - 1) - 3) + 3) - 2 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": "Find the total number of local minima and maxima for f(x) = (2(5((2x)/(3x)) - 2) - 1)/(1(5((2x)/(3x)) - 2) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 5(1cos(3\u03c0((-3x)/(-3x - 1)) + 2) + 1)^2 + 5(1cos(3\u03c0((-3x)/(-3x - 1)) + 2) + 1) - 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) = -3(1cos(1\u03c0(2|x + 3| + 1) - 2) - 1)^2 - 4(1cos(1\u03c0(2|x + 3| + 1) - 2) - 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": "Find the total number of local minima and maxima for f(x) = 5(-2sin(3\u03c0(-1x^2 - 4x - 4)) + 3)^2 - 1(-2sin(3\u03c0(-1x^2 - 4x - 4)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "541", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3e^(-1(-4(-1|x + 1| - 3) - 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": "Find the number of local maximum points for f(x) = (-2(2sin(2\u03c0(4x^2 + 5x) + 2)) - 1)/(-1(2sin(2\u03c0(4x^2 + 5x) + 2)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "395", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -3|(2sin(2\u03c0((-1x)/(-1x + 3))) - 1) + 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3e^(1(-1(-1x + 5)^2 - 4(-1x + 5) - 2)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2cos(3\u03c0(5(-4x^2 - 5x - 5) + 1) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "474", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -2(-4(-2sin(1\u03c0x - 1) + 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": "Determine the total count of local extrema points for the function f(x) = (-1(3sin(3\u03c0(4x - 1) - 1) - 2) - 3)/(1(3sin(3\u03c0(4x - 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 number of local maximum points for f(x) = 2(-2cos(3\u03c0(2|x + 2| - 3) + 1) + 2)^2 - 1(-2cos(3\u03c0(2|x + 2| - 3) + 1) + 2) - 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 number of local maximum points for f(x) = (-1(4(-1cos(2\u03c0x + 3) - 1) + 4) + 2)/(3(4(-1cos(2\u03c0x + 3) - 1) + 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": "How many local maxima does the function f(x) = 1(-1sin(1\u03c0(2|x + 3| + 1)) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = -2cos(2\u03c0(-3|(1|x - 3|) + 2| - 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": "Find the number of local maximum points for f(x) = -2cos(3\u03c0(-2(-1x^2 - 2x - 4) + 4) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "230", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3sin(3\u03c0(2|(1x - 4) - 1| - 3) - 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) = (2(3|(-2sin(2\u03c0x - 2) - 2) + 1|) + 2)/(2(3|(-2sin(2\u03c0x - 2) - 2) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3sin(3\u03c0(-2(-2x^2 + 2x) + 4) + 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3sin(2\u03c0((1(1x) + 2)/(2(1x) + 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": "Determine the count of local minima for the function f(x) = 2cos(2\u03c0((-2((-1x - 1)/(-2x + 3)))/(3((-1x - 1)/(-2x + 3)) - 1)) - 1) + 3 in the interval [-10, 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 minima for the function f(x) = -3|(-3(2cos(3\u03c0x) + 3)) - 3| + 1 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 extrema (minima and maxima combined) does the function f(x) = -1cos(2\u03c0(-3((-2x + 2)/(1x + 1)) - 4) + 1) + 2 have 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 number of local maximum points for f(x) = (-3(2cos(3\u03c0(-5x + 4) - 1) - 1) + 2)/(2(2cos(3\u03c0(-5x + 4) - 1) - 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "150", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 3cos(3\u03c0(-2|(5x^2 - 2x + 5) - 3| + 3)) - 2 have 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 total number of local minima and maxima for f(x) = 1sin(3\u03c0((-1(-5x^2 + 5x + 3) + 1)/(-1(-5x^2 + 5x + 3) - 2)) - 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) = 1cos(1\u03c0(4(2|x - 2| - 1)^2 - 1(2|x - 2| - 1) + 1) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 3cos(3\u03c0(-2|((-3x - 3)/(-1x - 1))| - 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = -5(-1cos(3\u03c0(1|x| - 1) - 3) - 2)^2 + 4(-1cos(3\u03c0(1|x| - 1) - 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": "Find the total number of local minima and maxima for f(x) = 2cos(1\u03c0(1(-1x + 4)^2 + 2) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2(-2sin(1\u03c0((-1x - 2)/(2x + 2))) + 3)^2 + 3 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(-2sin(1\u03c0(3x^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": "569", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = -1|((2(-2sin(2\u03c0x - 2) + 2))/(2(-2sin(2\u03c0x - 2) + 2) + 1)) + 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 maximum points for f(x) = -1cos(1\u03c0(-2|(-3|x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = (3(4(-3e^(1x) - 1)^2 + 1(-3e^(1x) - 1)))/(2(4(-3e^(1x) - 1)^2 + 1(-3e^(1x) - 1))) 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) = 2(2|(1sin(3\u03c0x + 2) - 2) + 1| + 3)^2 + 4 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) = 1|(-1|(1sin(3\u03c0x + 2) - 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": "Determine the count of local minima for the function f(x) = -2sin(3\u03c0(1(-1|x - 3| + 2)^2 + 1(-1|x - 3| + 2)) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "202", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3|((2(2sin(1\u03c0x - 3) - 1) + 2)/(-2(2sin(1\u03c0x - 3) - 1) - 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": "Determine the total count of local extrema points for the function f(x) = 3sin(3\u03c0(4(-5x - 5)^2 - 5(-5x - 5) + 3) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "346", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(-3(-3cos(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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = (2(2cos(1\u03c0(5x + 2) - 2) + 3) - 2)/(1(2cos(1\u03c0(5x + 2) - 2) + 3) + 2) 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) = 1sin(2\u03c0((-1(3x - 4) + 3)/(3(3x - 4) + 3)) + 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": "Determine the count of local maxima for the function f(x) = 3(2cos(2\u03c0((1x - 2)/(-2x)) + 1) + 3)^2 - 5(2cos(2\u03c0((1x - 2)/(-2x)) + 1) + 3) + 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": "Determine the count of local maxima for the function f(x) = 2|(-2|(-3cos(2\u03c0x + 3) - 1) + 2| + 1) - 1| + 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) = 4(-3cos(1\u03c0((3x - 1)/(1x + 1)) - 1) - 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) = -1|(-2(1cos(1\u03c0x + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 1|(1(-3sin(2\u03c0x - 3) + 3)^2 + 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": "Find the total number of local minima and maxima for f(x) = 1sin(2\u03c0((2(-3x) - 3)/(-3(-3x) + 2)) + 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 total number of local minima and maxima for f(x) = (-3(-3sin(2\u03c0(1x^2 - 3x - 5) + 2) - 2) - 2)/(3(-3sin(2\u03c0(1x^2 - 3x - 5) + 2) - 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "408", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -2sin(1\u03c0((-3(1|x - 1|) + 1)/(1(1|x - 1|))) + 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": "Determine the count of local maxima for the function f(x) = (-3(1sin(2\u03c0((-3x)/(-2x - 1)) - 3) - 1) + 2)/(-2(1sin(2\u03c0((-3x)/(-2x - 1)) - 3) - 1) + 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 count of local minima for the function f(x) = -2(-3cos(2\u03c0(2|x + 2| + 1) - 2) - 1)^2 - 4(-3cos(2\u03c0(2|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": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3sin(1\u03c0(-5(2|x + 1| + 1) - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2cos(1\u03c0(-4(-3x + 3)^2 + 1(-3x + 3) + 2)) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3(2|(3sin(2\u03c0x - 1)) - 2| + 3)^2 + 1(2|(3sin(2\u03c0x - 1)) - 2| + 3) - 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": "Find the number of local maximum points for f(x) = -2(-2cos(1\u03c0((-3x + 1)/(3x)) + 3) + 3) + 4 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": "How many local maxima does the function f(x) = 1|(2|(-3cos(2\u03c0x - 1) - 3) + 1| + 2) - 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -1((2(3cos(1\u03c0x + 2) - 1) - 2)/(-3(3cos(1\u03c0x + 2) - 1) - 2))^2 + 1((2(3cos(1\u03c0x + 2) - 1) - 2)/(-3(3cos(1\u03c0x + 2) - 1) - 2)) - 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 count of local maxima for the function f(x) = (2(-2e^(-2(1x^2 + 4x)) - 3) + 2)/(3(-2e^(-2(1x^2 + 4x)) - 3) - 2) 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 total count of local extrema points for the function f(x) = -2|(1cos(1\u03c0(-1|x - 3| - 2) + 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 maxima for the function f(x) = (-3(3cos(2\u03c0(2|x|) - 3) + 3) - 1)/(-2(3cos(2\u03c0(2|x|) - 3) + 3) + 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": "Determine the count of local maxima for the function f(x) = -2(-3|((2x + 3)/(-3x - 1)) - 1| + 3)^2 - 1(-3|((2x + 3)/(-3x - 1)) - 1| + 3) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2sin(1\u03c0(-3(3x + 5) + 2) - 3) + 3 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 maxima does the function f(x) = (-3(-2|(-3sin(1\u03c0x + 1) - 1) + 2| - 3) - 2)/(1(-2|(-3sin(1\u03c0x + 1) - 1) + 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": "Find the number of local maximum points for f(x) = 3(1|(-1sin(2\u03c0x - 3) - 2) - 3| + 3) + 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 minimum points for f(x) = -4(3sin(2\u03c0((-2x + 2)/(2x + 2)) + 2))^2 - 1(3sin(2\u03c0((-2x + 2)/(2x + 2)) + 2)) - 1 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": "Determine the total count of local extrema points for the function f(x) = 3|(-1cos(3\u03c0(2|x + 2| - 3) - 3) - 3)| in the interval [-10, 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 maxima for the function f(x) = 2sin(2\u03c0(2((2x + 2)/(-3x + 4))) + 1) + 1 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 extrema (minima and maxima combined) does the function f(x) = -3((-2(-1x^2 + 4x + 4) - 1)/(-2(-1x^2 + 4x + 4) + 3))^2 - 4((-2(-1x^2 + 4x + 4) - 1)/(-2(-1x^2 + 4x + 4) + 3)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = (3(2sin(3\u03c0(-1|x - 2| + 3) - 1)) - 2)/(-3(2sin(3\u03c0(-1|x - 2| + 3) - 1))) have 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) = 1(2cos(1\u03c0(-5x^2 - 4x - 2)) - 3)^2 + 2(2cos(1\u03c0(-5x^2 - 4x - 2)) - 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "496", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 1|(-3sin(1\u03c0(-2x + 1) - 3)) - 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": "Find the number of local maximum points for f(x) = -1(2sin(1\u03c0(-2x + 2) + 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": "Find the number of local minimum points for f(x) = -4(1|(3cos(1\u03c0x + 2) + 1) - 1| + 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": "How many local maxima does the function f(x) = -3(3cos(2\u03c0(5x - 1) + 3))^2 - 4(3cos(2\u03c0(5x - 1) + 3)) - 3 have 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) = 2cos(3\u03c0(-4(2x^2 + 1x) - 4) + 1) 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": "Determine the total count of local extrema points for the function f(x) = (-3(-2sin(2\u03c0(1x + 5) + 3) - 1) + 2)/(2(-2sin(2\u03c0(1x + 5) + 3) - 1) + 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": "How many local maxima does the function f(x) = -1cos(1\u03c0(1((-3x + 2)/(-2x - 1))^2 + 4) + 2) + 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": "Determine the count of local minima for the function f(x) = -2(-2cos(1\u03c0(-4x^2 - 1x + 4) - 3) + 3) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "289", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3|(2|(1x^2 + 2x) - 2| - 1) - 2| - 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": "Find the number of local minimum points for f(x) = -5(1sin(1\u03c0(-1|x - 1| + 2) - 1) - 1) + 5 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": "Find the number of local minimum points for f(x) = -1(3cos(1\u03c0(2x - 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) = 3|(3|(2cos(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": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 3|(-4(-1cos(1\u03c0x - 1) - 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": "Find the number of local minimum points for f(x) = -1(2sin(1\u03c0(-5x + 5) + 1) + 3)^2 - 5(2sin(1\u03c0(-5x + 5) + 1) + 3) + 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 count of local minima for the function f(x) = -1|(2sin(2\u03c0(1|x - 2| - 2) + 3) - 3) + 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 count of local maxima for the function f(x) = 3cos(3\u03c0(1|(5x^2 - 3x - 4) + 1| + 1) - 3) + 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 maxima does the function f(x) = 4(-2sin(2\u03c0(3|x + 3| + 1) + 1) + 3)^2 - 4(-2sin(2\u03c0(3|x + 3| + 1) + 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3sin(2\u03c0(-2(-3x + 2)^2 - 1(-3x + 2) - 4) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "488", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = -2((-3(-1sin(1\u03c0x + 3) - 3) - 1)/(-2(-1sin(1\u03c0x + 3) - 3) - 3))^2 - 5((-3(-1sin(1\u03c0x + 3) - 3) - 1)/(-2(-1sin(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": "How many local maxima does the function f(x) = -4(-1(-1sin(1\u03c0x - 3)) + 5) + 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": "Find the number of local maximum points for f(x) = -1|((1(3sin(2\u03c0x + 1) - 1))/(2(3sin(2\u03c0x + 1) - 1) - 1)) - 3| - 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) = 5((2(-3sin(2\u03c0x - 1) - 2) + 3)/(-3(-3sin(2\u03c0x - 1) - 2) - 1))^2 + 3((2(-3sin(2\u03c0x - 1) - 2) + 3)/(-3(-3sin(2\u03c0x - 1) - 2) - 1)) + 5 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 minima does the function f(x) = (-3(2|(1cos(2\u03c0x + 3)) - 1| - 1) + 2)/(1(2|(1cos(2\u03c0x + 3)) - 1| - 1) - 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) = (-1(2|(1sin(3\u03c0x + 1))| - 1) - 3)/(-2(2|(1sin(3\u03c0x + 1))| - 1)) 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) = -1|(1sin(1\u03c0(3x^2 - 5x - 4) - 2) - 2) + 2| - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "451", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1sin(3\u03c0(2(2|x + 2| - 3)) + 3) + 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 count of local maxima for the function f(x) = -2|(-1sin(3\u03c0(-3x + 3) + 3) + 2) + 1| + 3 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) = 2|((1(1cos(1\u03c0x + 1) + 2) + 3)/(-1(1cos(1\u03c0x + 1) + 2))) - 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 minima for the function f(x) = 3cos(2\u03c0(-3|(-2|x + 1| + 1) + 3| + 3) + 3) - 2 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 minima does the function f(x) = 1(-1(2cos(1\u03c0x - 1) - 1) - 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": "Find the number of local minimum points for f(x) = -3|(1cos(1\u03c0((2x + 3)/(1x + 2)) + 1) + 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": "Find the number of local minimum points for f(x) = -2((1(-1sin(1\u03c0x + 3) - 1) + 3)/(-2(-1sin(1\u03c0x + 3) - 1) - 1))^2 + 5((1(-1sin(1\u03c0x + 3) - 1) + 3)/(-2(-1sin(1\u03c0x + 3) - 1) - 1)) - 4 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) = -1|((-1(2sin(3\u03c0x + 2)) - 3)/(1(2sin(3\u03c0x + 2)) + 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": "How many local minima does the function f(x) = -3|(3sin(3\u03c0(-2x^2 - 2x + 5) - 1)) + 1| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "492", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -5(-1e^(2(-3|x + 3|)) + 2)^2 + 5(-1e^(2(-3|x + 3|)) + 2) - 5 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(1cos(1\u03c0x) + 1) + 4)^2 + 3(1(1cos(1\u03c0x) + 1) + 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 minima for the function f(x) = 4(3cos(1\u03c0(-4x^2 + 4) + 2) + 1)^2 + 1(3cos(1\u03c0(-4x^2 + 4) + 2) + 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2cos(2\u03c0(1(-1|x + 1| + 2) - 5) - 2) - 2 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) = 1|(-3(3cos(2\u03c0x + 1) + 3)^2 + 1(3cos(2\u03c0x + 1) + 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 count of local maxima for the function f(x) = -3((-2((-3x + 3)/(-2x + 3)) - 2)/(3((-3x + 3)/(-2x + 3)) + 3))^2 - 4((-2((-3x + 3)/(-2x + 3)) - 2)/(3((-3x + 3)/(-2x + 3)) + 3)) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "205", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3cos(3\u03c0(4(-4x - 4) + 2) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "959", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2|(-2cos(2\u03c0(-4x - 4) - 2) + 1)| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "320", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 3|(2cos(3\u03c0((2x + 1)/(-1x - 3))) + 2) + 3| + 2 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": "Find the number of local maximum points for f(x) = -2cos(3\u03c0(3(-1x + 3) + 3) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2sin(2\u03c0(1(1|x - 2| - 1)^2 - 1(1|x - 2| - 1) + 4) - 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 1(1sin(3\u03c0(2|x - 3| - 1) - 3) - 2) + 3 in the interval [-10, 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 maxima for the function f(x) = -1cos(2\u03c0(-2|(-3x^2 + 3x - 2) + 3| + 1) + 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "244", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2sin(2\u03c0((-2(1x + 3) + 1)/(-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": "72", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 2(1cos(2\u03c0(5x^2 + 5x + 3) - 2) - 2)^2 + 1(1cos(2\u03c0(5x^2 + 5x + 3) - 2) - 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "How many local minima does the function f(x) = 1sin(2\u03c0((2((-1x + 1)/(3x)) + 3)/(2((-1x + 1)/(3x)) + 2)) - 1) + 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 total number of local minima and maxima for f(x) = -2(2|(1cos(1\u03c0x + 1) - 1) - 3|)^2 - 4(2|(1cos(1\u03c0x + 1) - 1) - 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) = 1(1(1|x - 3| - 1)^2 - 5(1|x - 3| - 1) + 1)^2 + 4(1(1|x - 3| - 1)^2 - 5(1|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": "How many local maxima does the function f(x) = 3cos(1\u03c0(-1|(3x + 4) + 2| + 2)) + 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 minimum points for f(x) = 1(-1(-2cos(3\u03c0x + 3) - 3) + 2)^2 - 1(-1(-2cos(3\u03c0x + 3) - 3) + 2) - 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": "Determine the total count of local extrema points for the function f(x) = -3((1(1cos(3\u03c0x + 2)))/(-3(1cos(3\u03c0x + 2)) + 2)) - 5 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) = -1sin(3\u03c0(-3|(-3|x| - 3) - 1| + 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "541", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 2|(1sin(2\u03c0(-5x - 3) - 1) - 1) + 3| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -4(2|(-1sin(2\u03c0x - 1) + 2) + 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": "Find the total number of local minima and maxima for f(x) = -2sin(2\u03c0((-2(1|x| - 2) + 2)/(1(1|x| - 2) - 2)) - 3) + 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 count of local minima for the function f(x) = -1cos(3\u03c0(-2((1x - 3)/(1x + 1))^2 - 4((1x - 3)/(1x + 1)) - 3) + 1) + 3 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 minima for the function f(x) = -1|(1sin(3\u03c0((3x + 3)/(-2x - 1)) - 3) + 2) + 2| - 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 maxima does the function f(x) = -3sin(3\u03c0(-3|(4x - 2) - 1| + 3) - 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "359", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-2(3(-2cos(1\u03c0x + 2) - 3) - 1) + 1)/(3(3(-2cos(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": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-1(-3(-2cos(3\u03c0x - 1) - 1)^2 + 2(-2cos(3\u03c0x - 1) - 1) + 3) + 3)/(2(-3(-2cos(3\u03c0x - 1) - 1)^2 + 2(-2cos(3\u03c0x - 1) - 1) + 3) + 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 total count of local extrema points for the function f(x) = 4(-2|((2x - 1)/(-2x + 3)) + 3|)^2 + 2(-2|((2x - 1)/(-2x + 3)) + 3|) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2e^(-2(-1((-2x - 3)/(-1x - 1))^2 + 5((-2x - 3)/(-1x - 1)) + 5)) - 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 count of local minima for the function f(x) = 2(-1sin(2\u03c0((3x - 3)/(-2x - 1)) + 1) + 3)^2 - 5(-1sin(2\u03c0((3x - 3)/(-2x - 1)) + 1) + 3) + 4 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) = -3cos(3\u03c0((-1(5x^2 + 5x + 5) - 1)/(1(5x^2 + 5x + 5) - 3)) + 1) - 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": "Find the total number of local minima and maxima for f(x) = -2(-2sin(3\u03c0(3x - 5) - 3) + 3) 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 maximum points for f(x) = 5(-3cos(2\u03c0((-1x - 3)/(-1x - 2)) + 1) + 1) + 1 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) = 2|(-5(1|x - 1| - 2)^2 - 1(1|x - 1| - 2) + 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 number of local minimum points for f(x) = (-3(-3sin(3\u03c0((2x)/(-2x + 3)) + 2) + 1) + 2)/(2(-3sin(3\u03c0((2x)/(-2x + 3)) + 2) + 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": "Find the number of local maximum points for f(x) = (2(1cos(3\u03c0(-1x + 1) - 3)))/(-1(1cos(3\u03c0(-1x + 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": "Find the total number of local minima and maxima for f(x) = -3cos(1\u03c0(3(-2x^2 - 1x - 2)^2 - 2(-2x^2 - 1x - 2) + 5) - 3) + 1 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": "Find the number of local minimum points for f(x) = 1cos(3\u03c0(1|(1|x + 2| - 3) + 3|) + 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": "How many local minima does the function f(x) = -2cos(1\u03c0(-5(1x - 5)^2 - 3(1x - 5) - 3) - 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "248", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|((3(3cos(3\u03c0x + 3) + 3) + 3)/(-1(3cos(3\u03c0x + 3) + 3) - 1)) + 2| - 3 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 maxima for the function f(x) = -3sin(1\u03c0(-2(-4x^2 - 1x - 4)^2 - 4(-4x^2 - 1x - 4) + 3) + 3) + 3 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 maxima does the function f(x) = -3cos(2\u03c0((-2(-3|x - 2| + 2) - 2)/(-1(-3|x - 2| + 2) + 2)) + 3) + 1 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": "How many local minima does the function f(x) = 1(-2|(1cos(1\u03c0x - 3) - 3) + 3| - 1)^2 - 1(-2|(1cos(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": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (-2(2(1cos(2\u03c0x + 3) - 3)^2 - 1(1cos(2\u03c0x + 3) - 3) - 5) - 2)/(3(2(1cos(2\u03c0x + 3) - 3)^2 - 1(1cos(2\u03c0x + 3) - 3) - 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) = 1(-2|(-1sin(1\u03c0x - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = -2(-1cos(3\u03c0(5x + 2) - 1) + 1) - 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": "Find the number of local minimum points for f(x) = 1sin(1\u03c0(-5(-4x^2 + 4x + 1)^2 - 5(-4x^2 + 4x + 1) - 1) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5(-2cos(3\u03c0((2x - 1)/(-2x + 3)) + 1)) + 2 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 minima does the function f(x) = -1cos(2\u03c0((1((-3x)/(2x + 3)) + 1)/(1((-3x)/(2x + 3)) - 2))) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -3(-3cos(2\u03c0((1x + 3)/(3x + 2)) - 3) + 1)^2 + 1(-3cos(2\u03c0((1x + 3)/(3x + 2)) - 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": "Find the number of local maximum points for f(x) = 4(-3sin(2\u03c0(-4x^2 + 1) - 3) + 3)^2 + 2(-3sin(2\u03c0(-4x^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": "217", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 2sin(1\u03c0(1((-2x + 3)/(-3x + 1))^2 - 1((-2x + 3)/(-3x + 1)) - 4) + 2) + 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 count of local minima for the function f(x) = -1sin(2\u03c0((3(1x - 1) - 3)/(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": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = 5((-2(-1cos(1\u03c0x + 3) - 2) - 2)/(3(-1cos(1\u03c0x + 3) - 2) + 1)) - 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": "Find the number of local maximum points for f(x) = -1cos(1\u03c0(-1(-1x - 1)^2 + 5(-1x - 1) + 3) + 3) + 2 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": "Find the number of local maximum points for f(x) = 1(-1(-2cos(2\u03c0x - 1) + 2) + 5)^2 - 3(-1(-2cos(2\u03c0x - 1) + 2) + 5) + 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": "Find the number of local maximum points for f(x) = 1(-1sin(2\u03c0(-3x^2 - 5x - 1) - 2) - 3) - 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": "How many local maxima does the function f(x) = 2sin(3\u03c0(-2((1x - 3)/(-1x)) - 5) + 2) - 2 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 total number of local minima and maxima for f(x) = (3(1sin(2\u03c0(-3|x - 1| - 3) + 2) + 1) - 1)/(2(1sin(2\u03c0(-3|x - 1| - 3) + 2) + 1) - 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": "Determine the count of local maxima for the function f(x) = (-2(-2sin(2\u03c0(-5x - 2) - 2) - 2) - 1)/(-3(-2sin(2\u03c0(-5x - 2) - 2) - 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 total count of local extrema points for the function f(x) = 2(-2sin(3\u03c0(5x^2 - 1x + 5) - 1) - 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "494", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = (-2(-3cos(2\u03c0(3x^2 - 5x + 4) + 2)) - 3)/(2(-3cos(2\u03c0(3x^2 - 5x + 4) + 2)) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "228", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -1cos(2\u03c0(5((1x + 1)/(-2x - 3)) + 3) + 3) + 2 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 maxima does the function f(x) = (-2(-2sin(3\u03c0(2|x + 1| - 1) - 2) + 2) - 2)/(2(-2sin(3\u03c0(2|x + 1| - 1) - 2) + 2) + 3) 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) = 2sin(2\u03c0(1((-1x + 3)/(-2x + 3)) + 3) + 3) - 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": "Determine the count of local minima for the function f(x) = -3cos(1\u03c0(3(-3|x| - 1) + 4) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -1(-3(3sin(2\u03c0x))^2 + 2(3sin(2\u03c0x)) - 1)^2 - 4(-3(3sin(2\u03c0x))^2 + 2(3sin(2\u03c0x)) - 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2|(-1(-3cos(1\u03c0x + 1) - 2)^2 + 2(-3cos(1\u03c0x + 1) - 2) + 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 minima for the function f(x) = 4(-4(-1sin(2\u03c0x - 3) + 2)^2 - 2(-1sin(2\u03c0x - 3) + 2) + 4) + 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": "Determine the count of local maxima for the function f(x) = 1cos(3\u03c0(-2(1x - 5) - 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": "How many local minima does the function f(x) = 2(2cos(1\u03c0(-5x^2 + 2x + 5) + 3) + 2) + 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": "Find the number of local maximum points for f(x) = -1sin(2\u03c0(1|(3|x - 2| + 3) + 2| - 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": "Find the number of local minimum points for f(x) = (-2(-5(-2cos(2\u03c0x + 1) + 2) - 2) - 2)/(-2(-5(-2cos(2\u03c0x + 1) + 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": "How many local extrema (minima and maxima combined) does the function f(x) = 1(-2sin(2\u03c0(2x^2 - 1x - 4) - 1) - 1)^2 + 2(-2sin(2\u03c0(2x^2 - 1x - 4) - 1) - 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "525", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 2|(2e^(-1(5x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2(-2cos(1\u03c0(-3x + 5) - 3) + 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 maxima does the function f(x) = 2|(5(4x^2 - 5x + 1)^2 + 2(4x^2 - 5x + 1) - 5) + 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2sin(1\u03c0((-1((1x + 3)/(-2x - 1)) + 1)/(-3((1x + 3)/(-2x - 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": "Find the number of local maximum points for f(x) = 5(-2sin(1\u03c0(3|x + 3| + 1) + 1) - 2)^2 + 2(-2sin(1\u03c0(3|x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1cos(1\u03c0(2|((-1x + 1)/(1x)) - 3| - 2) - 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 minima for the function f(x) = 1(-2(1cos(3\u03c0x - 3) + 3)^2 - 4(1cos(3\u03c0x - 3) + 3) + 5)^2 + 1(-2(1cos(3\u03c0x - 3) + 3)^2 - 4(1cos(3\u03c0x - 3) + 3) + 5) - 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) = -1(-1(-3sin(1\u03c0x) + 1)^2 - 3(-3sin(1\u03c0x) + 1) - 5)^2 - 3(-1(-3sin(1\u03c0x) + 1)^2 - 3(-3sin(1\u03c0x) + 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 maximum points for f(x) = -3(3cos(2\u03c0(3x^2 - 5x - 2) + 1))^2 - 1(3cos(2\u03c0(3x^2 - 5x - 2) + 1)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "232", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -3cos(3\u03c0(-4(4x - 4) - 4) - 3) + 2 where -10 \u2264 x \u2264 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) = -2(-5(-3cos(3\u03c0x + 2) - 2)^2 + 5(-3cos(3\u03c0x + 2) - 2) + 2)^2 + 5(-5(-3cos(3\u03c0x + 2) - 2)^2 + 5(-3cos(3\u03c0x + 2) - 2) + 2) - 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": "Find the number of local minimum points for f(x) = -2sin(1\u03c0(1((2x + 3)/(1x - 2)) + 5) - 1) + 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": "How many local minima does the function f(x) = -3cos(2\u03c0((2(2|x - 1| - 1))/(3(2|x - 1| - 1) - 2))) - 3 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 maximum points for f(x) = -2|(2sin(3\u03c0(-4x^2 + 4x - 3) + 2) - 2) - 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "245", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = -2sin(2\u03c0(-1((3x + 1)/(-1x + 3))^2 + 2((3x + 1)/(-1x + 3)) + 4) + 2) + 3 where -10 \u2264 x \u2264 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(-2sin(2\u03c0((-1x + 1)/(-2x)) + 2) - 1) + 1)/(3(-2sin(2\u03c0((-1x + 1)/(-2x)) + 2) - 1) + 1) 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) = 3cos(3\u03c0((1((-2x + 2)/(1x + 2)))/(-2((-2x + 2)/(1x + 2)) - 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 maxima for the function f(x) = 2|((3(1sin(2\u03c0x - 1) - 2))/(3(1sin(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": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 5((-1(3sin(3\u03c0x + 2) - 1) - 1)/(-1(3sin(3\u03c0x + 2) - 1) - 2)) - 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 minima does the function f(x) = (-2(2|(-2sin(2\u03c0x + 1) - 1) - 1|) - 3)/(-2(2|(-2sin(2\u03c0x + 1) - 1) - 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(2|(3sin(2\u03c0x - 3) - 1) + 2| + 3) - 3)/(3(2|(3sin(2\u03c0x - 3) - 1) + 2| + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = (1(5(2cos(2\u03c0x) + 3)^2 + 2(2cos(2\u03c0x) + 3) - 2))/(3(5(2cos(2\u03c0x) + 3)^2 + 2(2cos(2\u03c0x) + 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 total count of local extrema points for the function f(x) = 2|(-2(2cos(3\u03c0x))^2 - 2(2cos(3\u03c0x)) - 3) + 2| + 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 count of local maxima for the function f(x) = (3(-2|(-1cos(3\u03c0x) + 2) + 3| + 2))/(-2(-2|(-1cos(3\u03c0x) + 2) + 3| + 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": "Find the number of local minimum points for f(x) = -1(3sin(1\u03c0(-4x) - 2))^2 - 2(3sin(1\u03c0(-4x) - 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 count of local maxima for the function f(x) = -1|(-4(-1sin(3\u03c0x + 3))^2 + 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 minima for the function f(x) = -1(-3(-3sin(1\u03c0x + 1) + 2) - 4)^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": "Find the number of local maximum points for f(x) = -1|(-1sin(3\u03c0((3x - 3)/(-2x + 3)) + 2) - 2) - 3| + 1 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(2e^(1(4x + 1)) - 1) - 2 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) = -4(2sin(1\u03c0(-2x^2 + 1x + 5) - 1) + 2) + 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": "Determine the count of local minima for the function f(x) = -2((-1(-2cos(1\u03c0x - 3) + 3) - 3)/(-2(-2cos(1\u03c0x - 3) + 3) + 3)) + 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": "Find the number of local maximum points for f(x) = -2(3sin(2\u03c0(-1|x - 3| + 2) + 3) - 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) = -1(-1|(3sin(2\u03c0x - 3) - 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) = -2(-5(3sin(1\u03c0x - 1) + 2)^2 + 3(3sin(1\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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 3((3((3x + 3)/(1x + 1)))/(-1((3x + 3)/(1x + 1)) - 1)) + 3 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": "Determine the count of local minima for the function f(x) = (-2(4(1sin(3\u03c0x - 3) + 3) + 2) - 2)/(1(4(1sin(3\u03c0x - 3) + 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 total number of local minima and maxima for f(x) = (2(1|(-1cos(3\u03c0x - 1)) - 1|))/(2(1|(-1cos(3\u03c0x - 1)) - 1|) - 1) where -10 \u2264 x \u2264 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 minimum points for f(x) = -1cos(1\u03c0(2((-3x)/(-2x - 3))) + 1) 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 number of local minimum points for f(x) = -1(-3|(1sin(2\u03c0x - 3) + 1) + 3| + 2)^2 - 1(-3|(1sin(2\u03c0x - 3) + 1) + 3| + 2) - 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": "How many local minima does the function f(x) = -1(-1sin(2\u03c0(1x^2 - 5x) + 2) + 2)^2 + 1(-1sin(2\u03c0(1x^2 - 5x) + 2) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "189", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = 5((-1(2cos(3\u03c0x - 1) - 1) + 3)/(1(2cos(3\u03c0x - 1) - 1)))^2 + 1((-1(2cos(3\u03c0x - 1) - 1) + 3)/(1(2cos(3\u03c0x - 1) - 1))) + 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 number of local maximum points for f(x) = -1sin(1\u03c0(5(4x + 1)^2 - 3(4x + 1) - 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "238", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = 3|((2(2sin(3\u03c0x + 1)) - 3)/(2(2sin(3\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": "179", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = 2(-1cos(3\u03c0((-2x + 2)/(2x + 1)) + 2) - 1) + 1 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 minima for the function f(x) = 5(3|(1sin(2\u03c0x - 3) + 2) + 1| - 2)^2 + 5(3|(1sin(2\u03c0x - 3) + 2) + 1| - 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 count of local minima for the function f(x) = -1(-3cos(1\u03c0(-4x + 1) - 1)) - 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": "Determine the total count of local extrema points for the function f(x) = -1(-1sin(2\u03c0(-5x^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": "499", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -3|(-1sin(1\u03c0(5x - 3) - 2) - 1) + 2| - 3 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) = 4(2sin(3\u03c0(3|x + 3| - 2)) - 3) - 5 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) = 1(1cos(3\u03c0(3x + 5)) + 2) + 4 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": "Determine the count of local maxima for the function f(x) = -3(-1cos(3\u03c0(1|x - 1| - 2) - 1) + 1) + 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) = -1sin(1\u03c0(-1(1x^2 + 5x - 1)) - 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -5(-2cos(1\u03c0(-4x - 5) + 2))^2 - 5(-2cos(1\u03c0(-4x - 5) + 2)) + 3 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) = -2(2(-1x^2 - 5x - 2)^2 - 5(-1x^2 - 5x - 2) - 1)^2 - 2(2(-1x^2 - 5x - 2)^2 - 5(-1x^2 - 5x - 2) - 1) - 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) = -1(3cos(3\u03c0(-2|x - 1| + 1) + 2) + 2)^2 + 2(3cos(3\u03c0(-2|x - 1| + 1) + 2) + 2) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "Determine the total count of local extrema points for the function f(x) = -3(-3cos(2\u03c0(1x - 1) - 2) + 2)^2 + 5(-3cos(2\u03c0(1x - 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": "How many local extrema (minima and maxima combined) does the function f(x) = -2(5(1sin(3\u03c0x) + 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": "How many local minima does the function f(x) = -5(2sin(3\u03c0(1x + 1) + 2) + 3)^2 + 1(2sin(3\u03c0(1x + 1) + 2) + 3) + 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": "Find the number of local maximum points for f(x) = -1(-3|(-3cos(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": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local minimum points for f(x) = 3((1(2cos(2\u03c0x) - 3) + 1)/(3(2cos(2\u03c0x) - 3) + 1)) - 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 total count of local extrema points for the function f(x) = 1|(1cos(2\u03c0(4x + 1) + 1) - 2) + 3| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 2sin(2\u03c0(3|(-5x^2 - 2x + 4) - 2|) - 1) + 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "248", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = -3sin(2\u03c0(1(-2|x - 3| - 1)^2 - 4(-2|x - 3| - 1) + 3) + 1) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "249", "dataset": "MATH"}
{"messages": [{"content": "How many local maxima does the function f(x) = 2|(1sin(2\u03c0(2|x + 1| + 1)) - 2)| - 1 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 minima does the function f(x) = 1sin(3\u03c0(3((-1x + 1)/(3x + 2)) + 2) - 2) - 3 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 total count of local extrema points for the function f(x) = (3(3(1sin(2\u03c0x + 1) + 2) + 4) + 3)/(-3(3(1sin(2\u03c0x + 1) + 2) + 4) + 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": "Find the total number of local minima and maxima for f(x) = -1sin(1\u03c0(-3|((3x)/(-3x + 1))| + 3) + 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 maximum points for f(x) = -3|(-3|(-1cos(2\u03c0x + 1) - 1) - 3|) - 1| + 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 total number of local minima and maxima for f(x) = 1cos(2\u03c0((-1(-1|x + 1|) + 2)/(1(-1|x + 1|) + 2)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "111", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local minima for the function f(x) = 3(-3cos(1\u03c0(3x^2 + 1x + 1) - 1) + 1)^2 - 3(-3cos(1\u03c0(3x^2 + 1x + 1) - 1) + 1) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "224", "dataset": "MATH"}
{"messages": [{"content": "Find the number of local maximum points for f(x) = (1(-1|(3cos(1\u03c0x - 3) + 1) - 2| + 3) + 3)/(1(-1|(3cos(1\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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1cos(3\u03c0(-3(-1x)^2 - 5(-1x) + 5) - 2) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "523", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2(-2sin(3\u03c0(-5x^2 + 4x - 4) - 1) - 1)^2 - 4(-2sin(3\u03c0(-5x^2 + 4x - 4) - 1) - 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": "How many local extrema (minima and maxima combined) does the function f(x) = 3(-4(-1x^2 - 5x - 3)^2 - 4(-1x^2 - 5x - 3) + 1)^2 - 3(-4(-1x^2 - 5x - 3)^2 - 4(-1x^2 - 5x - 3) + 1) - 2 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 count of local minima for the function f(x) = 3cos(1\u03c0(3|(1x^2 + 3x + 4) - 3|) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
{"messages": [{"content": "Find the total number of local minima and maxima for f(x) = (-2(-1|(3sin(3\u03c0x - 2) - 1) + 1| + 2) - 2)/(2(-1|(3sin(3\u03c0x - 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": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -1(-3sin(1\u03c0(3x^2 - 5x) - 1) - 1) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
{"messages": [{"content": "Determine the count of local maxima for the function f(x) = -2|(2e^(1(-3|x + 1|))) - 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) = -1(-1sin(3\u03c0(2|x| + 1) - 2) - 3)^2 + 1(-1sin(3\u03c0(2|x| + 1) - 2) - 3) - 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}