messages listlengths 1 1 | ground_truth stringlengths 1 3 | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = 4((2(3sin(1πx - 1) - 3) + 2)/(1(3sin(1πx - 1) - 3) - 2))^2 + 4((2(3sin(1πx - 1) - 3) + 2)/(1(3sin(1πx - 1) - 3) - 2)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 40 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -1|(2sin(2π((-3(4x + 5) - 2)/(-2(4x + 5) - 1))) + 1) - 1| + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1|(-1cos(2π((-1x - 3)/(-2x + 1)) - 2) + 3) - 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "How many local minima does the function f(x) = 3cos(2π(-1((-2(3x^2 - 5x + 1) - 2)/(-3(3x^2 - 5x + 1) + 4))) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 3|(3(-2sin(3πx + 3) + 3) + 2) + 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many local minima does the function f(x) = -1(-1sin(3π(5(3|x - 2| - 2)^2 - 3(3|x - 2| - 2) + 1)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 217 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -1(-1(2sin(1πx - 3) - 2) + 3)^2 - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = (3(3|(-3sin(2πx + 3) - 3) - 3| - 2) - 2)/(-3(3|(-3sin(2πx + 3) - 3) - 3| - 2) - 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 3((1(-2sin(1πx - 1) - 1) + 2)/(1(-2sin(1πx - 1) - 1)))^2 - 5((1(-2sin(1πx - 1) - 1) + 2)/(1(-2sin(1πx - 1) - 1))) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many local minima does the function f(x) = 2(1cos(3π(-2|(3x - 4) + 2|) - 3) + 1)^2 + 1(1cos(3π(-2|(3x - 4) + 2|) - 3) + 1) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 179 | MATH |
[
{
"content": "How many local minima does the function f(x) = (-1(4(1cos(3πx + 3) + 1) - 3) + 1)/(-1(4(1cos(3πx + 3) + 1) - 3) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -2|(-1(3sin(2π((-1x + 1)/(-2x + 3))) - 2) + 5) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = (1(3cos(2π(4x^2 + 2x - 2) + 3) + 2) - 2)/(-3(3cos(2π(4x^2 + 2x - 2) + 3) + 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 519 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = -2(-2sin(2π((-1(2|x - 3| + 3))/(-1(2|x - 3| + 3) + 2))) + 3) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 1|(2((2(2sin(1πx - 2) - 3))/(3(2sin(1πx - 2) - 3) + 1)) - 1) - 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = (-3(-2sin(3π(-5x + 2) + 3) - 2) - 1)/(-2(-2sin(3π(-5x + 2) + 3) - 2) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 300 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 1sin(1π(-1|(-5((-2x + 1)/(3x - 2))^2 + 5)| - 2) + 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 4(-2sin(1π((-3(-1|x| - 3) + 1)/(1(-1|x| - 3) + 3)) + 1)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 3cos(3π(-1|((-1(-5x^2 + 2x - 2) - 2)/(-1(-5x^2 + 2x - 2) - 3)) + 3| - 1)) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 1cos(1π(-5((2x - 2)/(-3x - 2))^2 + 1((2x - 2)/(-3x - 2)) - 5) + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 2sin(2π(-2(-1((-2x + 1)/(1x + 2)) + 3)^2 - 4(-1((-2x + 1)/(1x + 2)) + 3) + 1) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 140 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 3((-2(1cos(2π(5x - 4) - 1) + 3) + 2)/(2(1cos(2π(5x - 4) - 1) + 3) - 1))^2 - 5((-2(1cos(2π(5x - 4) - 1) + 3) + 2)/(2(1cos(2π(5x - 4) - 1) + 3) - 1)) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 100 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3((-1(1|(1cos(1πx - 3)) + 1| - 3) - 1)/(2(1|(1cos(1πx - 3)) + 1| - 3) - 1)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -2cos(3π((1(2x^2 - 5x - 1) + 2)/(-2(2x^2 - 5x - 1) - 1)) + 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = (2(-4(-1sin(2π(2|x + 2| + 2) - 1)) + 4))/(3(-4(-1sin(2π(2|x + 2| + 2) - 1)) + 4) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = (3(-4(1|x + 2| + 2) + 2))/(-2(-4(1|x + 2| + 2) + 2)) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 144 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2π((2(-1|(4x + 2) - 1| - 2) + 3)/(-1(-1|(4x + 2) - 1| - 2) + 3)) + 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = -2(1|(1sin(2π(-1x^2 + 3) + 3) - 1) + 3|) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 201 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 1|(-3(2cos(1πx + 1) - 2) - 5) + 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = 1(-2|(3cos(3π(-2x - 4) - 3) + 2) - 1| - 3)^2 - 5(-2|(3cos(3π(-2x - 4) - 3) + 2) - 1| - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 239 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -5(2sin(3π(-2|x + 3| - 1) - 1) + 2)^2 - 3(2sin(3π(-2|x + 3| - 1) - 1) + 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 121 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 2sin(1π(-2|(1x^2 + 4x - 2) - 2| - 2) - 1) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 209 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -4(-3|((-2(2sin(2πx + 1) - 2) + 2)/(2(2sin(2πx + 1) - 2) + 1)) + 1|)^2 + 4(-3|((-2(2sin(2πx + 1) - 2) + 2)/(2(2sin(2πx + 1) - 2) + 1)) + 1|) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 80 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -3(3|(-2cos(2π((2x - 3)/(-1x + 2)) - 2) + 2)| - 3)^2 - 4(3|(-2cos(2π((2x - 3)/(-1x + 2)) - 2) + 2)| - 3) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -1sin(1π(1|(-5x + 2) - 1|) + 2) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 101 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 1(3((-2(-2sin(1πx + 3)) - 3)/(2(-2sin(1πx + 3)) + 1))^2 + 2((-2(-2sin(1πx + 3)) - 3)/(2(-2sin(1πx + 3)) + 1)) + 5) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -2sin(2π(-2|(-4(3x^2 + 1x + 2) + 1) + 2| - 1) + 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 492 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = 2(3((-2(1cos(1πx) + 2) - 3)/(3(1cos(1πx) + 2) + 3)) - 4)^2 - 5(3((-2(1cos(1πx) + 2) - 3)/(3(1cos(1πx) + 2) + 3)) - 4) - 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 19 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = (2(-2cos(3π(-3x - 3) + 2) - 2) + 1)/(3(-2cos(3π(-3x - 3) + 2) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 90 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -3|(-3cos(1π(-4x + 5) + 3) + 2) + 2| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = 2((2(-1e^(2(2|x + 1| - 1)) - 3) + 3)/(1(-1e^(2(2|x + 1| - 1)) - 3) + 1)) - 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = 1(3cos(1π(2|(-1x) + 3| - 2)) - 1)^2 - 2(3cos(1π(2|(-1x) + 3| - 2)) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -5(-2|((-1(2sin(3πx) + 1) - 3)/(-3(2sin(3πx) + 1) - 3)) + 1|) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = 3(-3sin(1π(-3|x - 1| + 2)) + 3)^2 + 4(-3sin(1π(-3|x - 1| + 2)) + 3) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -4(3(1sin(3πx + 2) + 1)^2 + 5(1sin(3πx + 2) + 1) - 4) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = -2|(-4(3(3cos(1πx - 1) + 3))^2 + 4(3(3cos(1πx - 1) + 3)) + 1) + 1| - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 3|((-2(1cos(1πx + 2) + 2) + 1)/(3(1cos(1πx + 2) + 2) - 2)) + 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 2(-1(3cos(3π(2|x + 2| - 3) - 3) - 1)^2 + 5(3cos(3π(2|x + 2| - 3) - 3) - 1) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 1sin(1π(1(3|x| + 1) + 1) - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 5(3|(1cos(3πx + 3)) + 1| + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 5(-1sin(1π(-3x^2 + 3x - 3) - 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 284 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3sin(1π(4((-2x + 1)/(-1x)) - 4)) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 2|((1(-1sin(3π(5x^2 + 3x + 5) + 3) - 3) + 1)/(1(-1sin(3π(5x^2 + 3x + 5) + 3) - 3) + 2)) - 3| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 456 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -4((-3(1|(1cos(1πx + 2)) - 2| + 2) - 2)/(1(1|(1cos(1πx + 2)) - 2| + 2)))^2 - 2((-3(1|(1cos(1πx + 2)) - 2| + 2) - 2)/(1(1|(1cos(1πx + 2)) - 2| + 2))) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 10 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -2|(-1cos(2π(-3(-5x + 2)^2 + 5(-5x + 2)) + 3) + 2)| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 199 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = -1|((-2(2sin(1π(2x^2 + 4x + 5) + 1) - 2) + 3)/(2(2sin(1π(2x^2 + 4x + 5) + 1) - 2) + 1)) + 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 493 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -2sin(2π(2((-2(-2|x - 2| - 1) + 1)/(1(-2|x - 2| - 1) - 3))^2 + 3((-2(-2|x - 2| - 1) + 1)/(1(-2|x - 2| - 1) - 3)) + 4) - 3) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 9 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = (3(-2|(3sin(2πx + 2) + 2)| + 3) - 3)/(-1(-2|(3sin(2πx + 2) + 2)| + 3)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 5(5(2sin(3πx - 2) + 2)^2 + 2(2sin(3πx - 2) + 2) + 5) + 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = (-1(-3sin(3π(-1|x| - 1)) - 2) + 3)/(-1(-3sin(3π(-1|x| - 1)) - 2) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -2(2cos(2π(2|((2x + 1)/(-1x - 2)) + 1| + 2) + 3) + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 5((-2(-3sin(1πx + 1)) + 3)/(-3(-3sin(1πx + 1)) + 4))^2 - 1((-2(-3sin(1πx + 1)) + 3)/(-3(-3sin(1πx + 1)) + 4)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many local maxima does the function f(x) = (-3(-1(1|(-2x - 4) - 2| - 1)^2 + 1) - 3)/(2(-1(1|(-2x - 4) - 2| - 1)^2 + 1) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 166 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 2((-1(2sin(2πx) - 2) - 1)/(-2(2sin(2πx) - 2) + 1)) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = (1(4(1cos(3πx - 3) - 1)) + 2)/(1(4(1cos(3πx - 3) - 1)) + 3) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 2|(1sin(2π((-1x + 2)/(1x)) - 3) - 2) - 3| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 2cos(3π((1(-1x + 3))/(2(-1x + 3) - 1)) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = (2(-1(2cos(2πx + 2) + 2)^2 + 1(2cos(2πx + 2) + 2) - 3) + 1)/(-1(-1(2cos(2πx + 2) + 2)^2 + 1(2cos(2πx + 2) + 2) - 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 2|(-3(1sin(2π(4x^2 - 5x + 4) - 3) - 1) + 4) - 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 1cos(3π(-5((-2x - 1)/(-2x + 3)) + 5) + 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 5((2(-1sin(3πx + 1) + 2) - 1)/(1(-1sin(3πx + 1) + 2) + 1))^2 + 3((2(-1sin(3πx + 1) + 2) - 1)/(1(-1sin(3πx + 1) + 2) + 1)) - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 4(-2(2sin(2π(-1|x - 2| + 3) - 3) + 3) + 2)^2 - 1(-2(2sin(2π(-1|x - 2| + 3) - 3) + 3) + 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = -3(-1sin(3π((-1x + 3)/(2x + 2)) - 2) - 2) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = 3(-2cos(2π((2(1|x - 2| - 2) + 2)/(1(1|x - 2| - 2))) - 3) + 1)^2 - 5(-2cos(2π((2(1|x - 2| - 2) + 2)/(1(1|x - 2| - 2))) - 3) + 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 111 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 3|(-4(4(-1cos(3πx + 2)) - 5)^2 - 1(4(-1cos(3πx + 2)) - 5) - 1) - 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = (-3(3(2cos(1π(4x + 2) - 1) - 2)^2 + 4(2cos(1π(4x + 2) - 1) - 2) - 1) + 2)/(-3(3(2cos(1π(4x + 2) - 1) - 2)^2 + 4(2cos(1π(4x + 2) - 1) - 2) - 1) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 80 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -1cos(1π((3(-3x + 5) - 2)/(-1(-3x + 5) - 3)) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 4(3(-3|((-3x - 3)/(2x + 2)) + 3|)^2 + 3(-3|((-3x - 3)/(2x + 2)) + 3|) - 4) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = 3sin(2π(-3((-1(-3x^2 - 2x) - 3)/(1(-3x^2 - 2x))) - 4) + 1) - 2 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -2(-5((1(-1cos(2πx - 1) + 3) + 1)/(-3(-1cos(2πx - 1) + 3) - 2)) - 1)^2 - 5(-5((1(-1cos(2πx - 1) + 3) + 1)/(-3(-1cos(2πx - 1) + 3) - 2)) - 1) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 20 | MATH |
[
{
"content": "Determine the count of local minima for the function f(x) = 3|(2sin(1π((3x - 3)/(2x + 3)) + 3) - 2)| in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = -3(-2cos(3π(3|x - 2| - 3) - 1) - 1)^2 + 3(-2cos(3π(3|x - 2| - 3) - 1) - 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 178 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = (-2(-2(-2cos(3πx - 2) - 2) + 3) + 1)/(3(-2(-2cos(3πx - 2) - 2) + 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 3|(1sin(2π(4(-1x^2 + 5x) - 4) + 1) + 1) + 2| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 479 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = -2cos(2π((-2(-2(-3x^2 + 5x + 3) - 2))/(1(-2(-3x^2 + 5x + 3) - 2) + 3)) - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Determine the count of local maxima for the function f(x) = (3(3cos(2π(2|(-2x^2 + 5)| - 2) + 1) - 1) - 1)/(-2(3cos(2π(2|(-2x^2 + 5)| - 2) + 1) - 1) + 1) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 218 | MATH |
[
{
"content": "Find the number of local maximum points for f(x) = 4(3sin(3π((2x - 2)/(1x - 2))) - 2)^2 - 2(3sin(3π((2x - 2)/(1x - 2))) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 1cos(1π(-4(2((-1x + 3)/(1x - 2)) + 2)^2 + 4(2((-1x + 3)/(1x - 2)) + 2) - 4) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -2sin(3π(-2(1((1x)/(3x + 3))^2 - 3) + 2) - 3) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many local minima does the function f(x) = -5(-1sin(3π((1x + 1)/(-1x + 2)) - 1) - 2) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 3((-2(-3|(-3cos(1πx + 2) + 3) - 1| - 1))/(1(-3|(-3cos(1πx + 2) + 3) - 1| - 1) + 2)) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many local extrema (minima and maxima combined) does the function f(x) = -3|(2cos(3π((2(1x - 5) + 2)/(-2(1x - 5) + 3)) - 3) - 2) + 3| + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "Find the total number of local minima and maxima for f(x) = 3(1|(-3cos(1πx) + 3) - 3| + 2)^2 - 5(1|(-3cos(1πx) + 3) - 3| + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = -1(-2(2cos(2π(-3|x - 2| - 3) - 1) + 2)^2 - 3(2cos(2π(-3|x - 2| - 3) - 1) + 2) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 2sin(2π(-3(1|(2x^2 - 4x + 1) - 1| - 3) - 3) + 2) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 250 | MATH |
[
{
"content": "How many local maxima does the function f(x) = (-2(1(-2(1cos(2πx) + 3) + 5)^2 + 2(-2(1cos(2πx) + 3) + 5) + 2) + 2)/(1(1(-2(1cos(2πx) + 3) + 5)^2 + 2(-2(1cos(2πx) + 3) + 5) + 2) + 3) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 40 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -2e^(2(-1|(-4x^2 + 1x + 1)|)) + 3 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -1sin(3π(-4(1|x + 3| - 3) - 1) + 2) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 119 | MATH |
[
{
"content": "Determine the total count of local extrema points for the function f(x) = -5((1(1sin(2π(3|x - 3| - 3) - 2) + 3) + 3)/(-2(1sin(2π(3|x - 3| - 3) - 2) + 3) - 2)) + 4 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 121 | MATH |
[
{
"content": "Find the number of local minimum points for f(x) = 2|((1(2sin(1πx + 3) - 1) - 2)/(-2(2sin(1πx + 3) - 1) + 3)) - 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
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