messages listlengths 1 1 | ground_truth stringclasses 42 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many local minima does the function f(x) = 2|(-3((1(2sin(3πx - 1) + 2))/(-2(2sin(3πx - 1) + 2) - 1))^2 + 5((1(2sin(3πx - 1) + 2))/(-2(2sin(3πx - 1) + 2) - 1)) + 2)| + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -2(-1|(-1cos(3π((2x + 3)/(3x + 3)) + 2) - 1) - 2| + 3)^2 - 2(-1|(-1cos(3π((2x + 3)/(3x + 3)) + 2) - 1) - 2| + 3) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Y... | -4 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local minimum of the function f(x) = -1cos(1π(-1|((3(1x^2 - 3x + 4))/(1(1x^2 - 3x + 4) - 2)) + 2|) + 2) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local minimum of the function f(x) = 1sin(2π(5(-3|((3x - 2)/(-1x - 2)) + 3| - 3)^2 - 2(-3|((3x - 2)/(-1x - 2)) + 3| - 3) + 1) - 2) - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local minimum of the function f(x) = (1(2|(1sin(3π(5x^2 + 3x - 2) - 2)) - 1| - 3) - 1)/(2(2|(1sin(3π(5x^2 + 3x - 2) - 2)) - 1| - 3) + 2) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the smallest y-value of the function f(x) = -1|(3(-2cos(3π((-2x)/(-3x + 4)) - 1))^2 + 4(-2cos(3π((-2x)/(-3x + 4)) - 1)) - 3) - 2| - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = 1sin(2π(-1|(5((1x - 3)/(2x - 3))^2 - 4((1x - 3)/(2x - 3))) + 3|) + 2) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = 1((2(3|(1sin(1πx + 3) + 2) + 1| + 2))/(3(3|(1sin(1πx + 3) + 2) + 1| + 2) - 1))^2 - 2((2(3|(1sin(1πx + 3) + 2) + 1| + 2))/(3(3|(1sin(1πx + 3) + 2) + 1| + 2) - 1)) + 2 in the interval [-10, 10]. Round to the nearest integer.... | -9 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local minimum of the function f(x) = -3(-1cos(3π(1|((2x - 1)/(3x + 2)) - 2| + 1)) - 2)^2 + 5(-1cos(3π(1|((2x - 1)/(3x + 2)) - 2| + 1)) - 2) + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 2 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the largest y-value of the function f(x) = (2(-2sin(3π(1|(-5x^2 + 3x)| - 1) + 2)) + 2)/(3(-2sin(3π(1|(-5x^2 + 3x)| - 1) + 2)) + 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = 1|((-1(4(-1sin(2πx) - 2)^2 + 2) - 3)/(-3(4(-1sin(2πx) - 2)^2 + 2) + 2)) - 3| + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | -9 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the largest y-value of the function f(x) = (3(-5(1|(2cos(1πx + 3)) - 3| - 2)^2 + 4(1|(2cos(1πx + 3)) - 3| - 2) - 1) + 3)/(1(-5(1|(2cos(1πx + 3)) - 3| - 2)^2 + 4(1|(2cos(1πx + 3)) - 3| - 2) - 1) - 3) in the interval [-10, 10]. Round to the nearest in... | -10 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -2|(4(3cos(3π((-1x - 3)/(2x + 3)) - 3) - 2)^2 + 3(3cos(3π((-1x - 3)/(2x + 3)) - 3) - 2) - 3) + 2| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = (-1(2sin(1π(-3|(3x^2 - 3x - 1) + 3| - 2)) + 1) + 2)/(-1(2sin(1π(-3|(3x^2 - 3x - 1) + 3| - 2)) + 1)) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{You... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum of the function f(x) = 1sin(1π((3(3|(4x^2 + 5x + 1) + 2|) - 1)/(-1(3|(4x^2 + 5x + 1) + 2|) - 3)) + 2) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = 1|((1(-1(1cos(1πx) + 2)^2 - 4(1cos(1πx) + 2) - 4) + 1)/(3(-1(1cos(1πx) + 2)^2 - 4(1cos(1πx) + 2) - 4) - 1)) - 2| - 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex fo... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = -2|((2(-5(-2cos(2πx + 3))^2 + 2(-2cos(2πx + 3)) - 3))/(-1(-5(-2cos(2πx + 3))^2 + 2(-2cos(2πx + 3)) - 3) - 2)) - 3| - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = 1cos(2π(-3|(5((1x - 1)/(1x))^2 + 5((1x - 1)/(1x)) - 4) + 1| + 1) + 3) - 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -9 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = 2((-1(-2sin(3π(-2|x - 2| + 3) + 1) + 1))/(-2(-2sin(3π(-2|x - 2| + 3) + 1) + 1) - 1))^2 + 4((-1(-2sin(3π(-2|x - 2| + 3) + 1) + 1))/(-2(-2sin(3π(-2|x - 2| + 3) + 1) + 1) - 1)) + 4 in the interval [-10, 10]. Round to the nea... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local extremum of the function f(x) = -2|(4(3cos(2π((3x - 3)/(-2x)) - 1) - 2)^2 + 4(3cos(2π((3x - 3)/(-2x)) - 1) - 2) - 5) - 1| + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 9 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = (-3(-3|(-3sin(3π(3x^2 - 5x) - 1)) + 3| + 3) - 1)/(-2(-3|(-3sin(3π(3x^2 - 5x) - 1)) + 3| + 3) - 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the largest y-value of the function f(x) = (1(-2(-1|(-3cos(2πx + 3) + 3) - 3| + 3)^2 + 3(-1|(-3cos(2πx + 3) + 3) - 3| + 3) - 4) - 3)/(3(-2(-1|(-3cos(2πx + 3) + 3) - 3| + 3)^2 + 3(-1|(-3cos(2πx + 3) + 3) - 3| + 3) - 4) - 1) in the interval [-10, 10].... | -10 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = -1cos(3π(3|(5((1x + 1)/(3x + 1))^2 - 1((1x + 1)/(3x + 1)) - 3) + 1| + 2) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = (-2(2cos(3π(2|(-1x^2 + 4x + 4) - 2| + 2) - 1) + 3) + 2)/(2(2cos(3π(2|(-1x^2 + 4x + 4) - 2| + 2) - 1) + 3) - 3) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: ... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -1|(1(-1cos(1π((-1x + 3)/(2x + 3)) - 1))^2 + 2(-1cos(1π((-1x + 3)/(2x + 3)) - 1)) - 4) + 3| in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | -3 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local maximum of the function f(x) = 1((1(-1cos(3π(1|x - 2| - 3)) + 3) - 3)/(2(-1cos(3π(1|x - 2| - 3)) + 3) - 3))^2 - 2((1(-1cos(3π(1|x - 2| - 3)) + 3) - 3)/(2(-1cos(3π(1|x - 2| - 3)) + 3) - 3)) - 4 in the interval [-10, 10]. Round to the nearest integer.\n\n... | 10 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the smallest y-value of the function f(x) = 1cos(3π(3(-2|((3x + 2)/(3x - 2))| - 2)^2 - 1(-2|((3x + 2)/(3x - 2))| - 2))) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | -9 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = 1cos(2π((3(2(-1|x + 2| + 1)^2 + 5) - 3)/(-1(2(-1|x + 2| + 1)^2 + 5) - 2)) - 1) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = -3sin(3π(-3|((-2(-4x^2 + 1) + 1)/(1(-4x^2 + 1) + 2)) - 3| - 3) - 1) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = -3sin(3π(-1|(5((-3x + 2)/(-2x + 3))^2 + 1((-3x + 2)/(-2x + 3)) - 5)| + 2) - 2) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | -7 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = (1(-3|(-1(3sin(3πx + 3) + 1)^2 + 4(3sin(3πx + 3) + 1) - 5)| - 1) + 2)/(3(-3|(-1(3sin(3πx + 3) + 1)^2 + 4(3sin(3πx + 3) + 1) - 5)| - 1) - 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = (3(-4(-2sin(2π(3|x - 3| - 3) + 1))^2 + 4(-2sin(2π(3|x - 3| - 3) + 1)) + 5))/(3(-4(-2sin(2π(3|x - 3| - 3) + 1))^2 + 4(-2sin(2π(3|x - 3| - 3) + 1)) + 5) + 1) in the interval [-10, 10]. Round to the nearest i... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = -1(-1sin(3π((-1(-3|x + 2| + 1) + 1)/(-2(-3|x + 2| + 1) - 2)) - 3) + 3)^2 + 3(-1sin(3π((-1(-3|x + 2| + 1) + 1)/(-2(-3|x + 2| + 1) - 2)) - 3) + 3) - 2 in the interval [-10, 10]. Round to the nearest integer.\n... | -4 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local minimum of the function f(x) = 2|(3(-1cos(1π((-1x - 3)/(-1x + 2)) - 3) - 3)^2 + 4(-1cos(1π((-1x - 3)/(-1x + 2)) - 3) - 3)) + 2| in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 7 | MATH |
[
{
"content": "How many local maxima does the function f(x) = (2(-2sin(3π(2(-2|x - 2| + 1)^2 - 5(-2|x - 2| + 1) - 3))) - 1)/(-1(-2sin(3π(2(-2|x - 2| + 1)^2 - 5(-2|x - 2| + 1) - 3))) + 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local minimum of the function f(x) = 1|(1(2cos(2π((-1x + 1)/(-3x + 2)) + 3) + 3)^2 + 5(2cos(2π((-1x + 1)/(-3x + 2)) + 3) + 3) - 1) + 3| + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = (-1(2(3|(1sin(1πx + 2) + 2)| + 3)^2 - 2(3|(1sin(1πx + 2) + 2)| + 3) + 2) + 1)/(3(2(3|(1sin(1πx + 2) + 2)| + 3)^2 - 2(3|(1sin(1πx + 2) + 2)| + 3) + 2) + 2) in the interval [-10, 10]. Round to the nearest int... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = 1|(1(-2sin(3π((-2x - 1)/(1x + 2)) - 3) - 2)^2 + 1(-2sin(3π((-2x - 1)/(1x + 2)) - 3) - 2) - 3)| + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | -9 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = 3cos(2π(2((-2(2|x + 3|))/(-3(2|x + 3|) - 2))^2 - 5((-2(2|x + 3|))/(-3(2|x + 3|) - 2)) - 5) + 1) + 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = -2(-3cos(1π((1(1|x - 3| - 2))/(-2(1|x - 3| - 2) + 3)) - 1) - 3)^2 + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = -3(-2sin(2π(-2|((-3x + 2)/(2x - 1)) - 3| - 3) + 2) + 3)^2 + 5(-2sin(2π(-2|((-3x + 2)/(2x - 1)) - 3| - 3) + 2) + 3) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex ... | -7 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local extremum of the function f(x) = 3|((-3(-3(2cos(1πx - 1) + 1)^2 + 5(2cos(1πx - 1) + 1) - 5))/(1(-3(2cos(1πx - 1) + 1)^2 + 5(2cos(1πx - 1) + 1) - 5))) + 2| - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format:... | 10 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = 2(3cos(2π((-1(1|x + 1| - 1) - 1)/(-1(1|x + 1| - 1) + 2)) + 1) - 3)^2 - 5(3cos(2π((-1(1|x + 1| - 1) - 1)/(-1(1|x + 1| - 1) + 2)) + 1) - 3) - 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPrese... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = 1sin(1π((3(3(3|x + 3| - 1)^2 - 3(3|x + 3| - 1)) - 3)/(2(3(3|x + 3| - 1)^2 - 3(3|x + 3| - 1)) - 3)) + 2) - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \... | -4 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = 1(3sin(1π(-1|((-3x - 1)/(1x + 3)) - 3| - 1) + 1))^2 - 5(3sin(1π(-1|((-3x - 1)/(1x + 3)) - 3| - 1) + 1)) + 4 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\b... | -7 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = -4((-1(1sin(2π(2|x - 1| + 1)) + 2) - 3)/(2(1sin(2π(2|x - 1| + 1)) + 2) - 3))^2 + 3((-1(1sin(2π(2|x - 1| + 1)) + 2) - 3)/(2(1sin(2π(2|x - 1| + 1)) + 2) - 3)) - 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex f... | 161 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = (-2(-2|(3(3sin(2πx - 1) + 3)^2 + 3(3sin(2πx - 1) + 3) - 5) - 2| + 1) - 1)/(1(-2|(3(3sin(2πx - 1) + 3)^2 + 3(3sin(2πx - 1) + 3) - 5) - 2| + 1)) in the interval [-10, 10]. Round to the nearest integer.\n\nPres... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = -2cos(1π(1(2|((2x + 2)/(2x - 1)) - 3| + 1)^2 + 3(2|((2x + 2)/(2x - 1)) - 3| + 1) + 5) + 1) - 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | -7 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the smallest y-value of the function f(x) = (-1(2|(1(2cos(1πx - 3) - 2)^2 + 5(2cos(1πx - 3) - 2) + 1) - 1| + 1) + 1)/(2(2|(1(2cos(1πx - 3) - 2)^2 + 5(2cos(1πx - 3) - 2) + 1) - 1| + 1) + 2) in the interval [-10, 10]. Round to the nearest integer.\n\nP... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = -4(-2|((1(3cos(1πx) - 1) + 3)/(1(3cos(1πx) - 1) + 1)) - 1| + 2)^2 + 3(-2|((1(3cos(1πx) - 1) + 3)/(1(3cos(1πx) - 1) + 1)) - 1| + 2) - 4 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the... | -9 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = (-1(-2(2cos(2π(-2|x - 1| - 1) - 3) - 3)^2 + 2(2cos(2π(-2|x - 1| - 1) - 3) - 3) - 1) - 1)/(-2(-2(2cos(2π(-2|x - 1| - 1) - 3) - 3)^2 + 2(2cos(2π(-2|x - 1| - 1) - 3) - 3) - 1) + 3) in the interval [-10, 10]. Round to the near... | -10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = 5(-3sin(1π((3(-2|x - 1| + 2))/(-3(-2|x - 1| + 2) - 1)) - 1) + 2)^2 + 5(-3sin(1π((3(-2|x - 1| + 2))/(-3(-2|x - 1| + 2) - 1)) - 1) + 2) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in ... | 1 | MATH |
[
{
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = 1sin(1π((3(-5(-3|x| + 2)^2 + 4(-3|x| + 2) + 3) + 3)/(-1(-5(-3|x| + 2)^2 + 4(-3|x| + 2) + 3) + 3)) + 1) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 0 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = -1|(-1((2(2sin(2πx) - 2) + 3)/(3(2sin(2πx) - 2) + 3))^2 - 1((2(2sin(2πx) - 2) + 3)/(3(2sin(2πx) - 2) + 3)) - 5) - 2| + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\bo... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local extremum of the function f(x) = -1sin(1π(3|((3(-3x^2 - 1x + 1) + 1)/(3(-3x^2 - 1x + 1))) - 3| - 3) + 1) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = (2(4(2cos(2π(-2|x| - 2) + 3) + 3)^2 - 4(2cos(2π(-2|x| - 2) + 3) + 3) + 5) - 2)/(3(4(2cos(2π(-2|x| - 2) + 3) + 3)^2 - 4(2cos(2π(-2|x| - 2) + 3) + 3) + 5)) in the interval [-10, 10]. Round to the nearest integ... | -10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = (-1(-1|(1cos(1π(-1x^2 - 3x - 4) + 2) - 1) - 3| - 1) - 3)/(2(-1|(1cos(1π(-1x^2 - 3x - 4) + 2) - 1) - 3| - 1) + 3) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your... | 1 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -3sin(2π(3(-1|((-2x - 2)/(-3x - 3)) + 1| - 3)^2 - 5) + 1) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -4 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the smallest y-value of the function f(x) = 2|(-5((-1(-1sin(3πx + 1) + 1) + 2)/(-3(-1sin(3πx + 1) + 1) - 2))^2 + 1((-1(-1sin(3πx + 1) + 1) + 2)/(-3(-1sin(3πx + 1) + 1) - 2)) + 4) + 2| + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPr... | -10 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = 5(2cos(2π((-1(3|x + 2| + 2) + 2)/(-2(3|x + 2| + 2) + 1)) - 2) - 1)^2 - 1(2cos(2π((-1(3|x + 2| + 2) + 2)/(-2(3|x + 2| + 2) + 1)) - 2) - 1) + 4 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your... | 6 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = -1sin(2π((1(-2|(-1x^2 + 2x + 4) - 1|) + 3)/(2(-2|(-1x^2 + 2x + 4) - 1|) - 1)) + 3) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | -2 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local minimum of the function f(x) = 4(3|(-3cos(3π((3x + 2)/(3x)) - 3) - 1) - 2| + 1)^2 - 3(3|(-3cos(3π((3x + 2)/(3x)) - 3) - 1) - 2| + 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | -2 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = (-3(-4(1sin(1π(-3|x + 3| + 2) - 3) - 1)^2 - 4(1sin(1π(-3|x + 3| + 2) - 3) - 1) + 2) - 2)/(-2(-4(1sin(1π(-3|x + 3| + 2) - 3) - 1)^2 - 4(1sin(1π(-3|x + 3| + 2) - 3) - 1) + 2) - 3) in the interval [-10, 10]. Round to the nea... | 0 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = 1(-3|((-3(-1sin(3πx - 1) + 3) + 3)/(-3(-1sin(3πx - 1) + 3) - 1)) - 3| - 1)^2 + 1(-3|((-3(-1sin(3πx - 1) + 3) + 3)/(-3(-1sin(3πx - 1) + 3) - 1)) - 3| - 1) + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPre... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the largest y-value of the function f(x) = (2(2|(-5(-1cos(3πx - 1) - 3)^2 + 2(-1cos(3πx - 1) - 3) - 1) - 3|) - 2)/(1(2|(-5(-1cos(3πx - 1) - 3)^2 + 2(-1cos(3πx - 1) - 3) - 1) - 3|)) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent ... | -10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = (-2(3cos(2π(-5(1|x + 2| - 1)^2 - 2(1|x + 2| - 1) - 2) + 2) - 3) + 2)/(-3(3cos(2π(-5(1|x + 2| - 1)^2 - 2(1|x + 2| - 1) - 2) + 2) - 3) - 2) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in ... | 0 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = 1|(1((2(3sin(1πx - 2)) - 1)/(1(3sin(1πx - 2)) + 1))^2 - 2((2(3sin(1πx - 2)) - 1)/(1(3sin(1πx - 2)) + 1)) + 3) - 2| + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxe... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local minimum of the function f(x) = (-3(1sin(1π(5(-3|x - 1| - 2)^2 - 2(-3|x - 1| - 2) + 5) + 2) + 2) + 3)/(2(1sin(1π(5(-3|x - 1| - 2)^2 - 2(-3|x - 1| - 2) + 5) + 2) + 2) + 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in L... | 10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = -4(-2|((-2(-2sin(3πx + 2) + 3) + 3)/(2(-2sin(3πx + 2) + 3) - 1)) + 2| - 3)^2 + 4(-2|((-2(-2sin(3πx + 2) + 3) + 3)/(2(-2sin(3πx + 2) + 3) - 1)) + 2| - 3) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent ... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local minimum of the function f(x) = 1((2(-2cos(1π(-2|x| - 3) + 3)) - 1)/(2(-2cos(1π(-2|x| - 3) + 3)) - 3))^2 + 3((2(-2cos(1π(-2|x| - 3) + 3)) - 1)/(2(-2cos(1π(-2|x| - 3) + 3)) - 3)) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answ... | 10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = -1sin(1π(4((-3(3|x - 3| - 3) - 3)/(-2(3|x - 3| - 3) - 2))^2 + 4) + 3) + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = (-2(-1sin(2π(3|(4x^2 - 2x) - 3| + 2) - 2) + 1) + 2)/(-1(-1sin(2π(3|(4x^2 - 2x) - 3| + 2) - 2) + 1) + 3) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = 3|(4(3cos(2π((-1x + 2)/(-3x)) - 3) - 1)^2 - 1(3cos(2π((-1x + 2)/(-3x)) - 3) - 1) - 2) + 1| - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | -5 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -2(1sin(1π((3(-3|x|) + 1)/(-3(-3|x|) + 2))) - 3)^2 + 2(1sin(1π((3(-3|x|) + 1)/(-3(-3|x|) + 2))) - 3) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the smallest y-value of the function f(x) = (-3(-2sin(3π(-2|(-3x^2 + 4x - 4) + 3|) + 2) - 3) + 1)/(1(-2sin(3π(-2|(-3x^2 + 4x - 4) + 3|) + 2) - 3) + 2) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\b... | 1 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = 2|(-5(-2sin(1π((1x - 1)/(1x + 1))) - 1)^2 + 4(-2sin(1π((1x - 1)/(1x + 1))) - 1) - 1)| - 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | -1 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -1|(-1((-3(3sin(3πx - 2)) + 3)/(2(3sin(3πx - 2)) - 1))^2 + 1((-3(3sin(3πx - 2)) + 3)/(2(3sin(3πx - 2)) - 1)) - 3) + 3| + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\b... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the rightmost local extremum of the function f(x) = 1(1|((-1(-1cos(3πx - 1) - 2) + 2)/(-3(-1cos(3πx - 1) - 2) - 2))| + 2)^2 - 1(1|((-1(-1cos(3πx - 1) - 2) + 2)/(-3(-1cos(3πx - 1) - 2) - 2))| + 2) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the a... | 10 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = -2|(-3sin(3π((2(5x^2 + 3x + 5) - 1)/(-2(5x^2 + 3x + 5) - 2)))) + 3| - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | -1 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = 1(3cos(2π(-3|((-2x)/(-3x - 1)) + 1| + 3)) - 3)^2 + 4(3cos(2π(-3|((-2x)/(-3x - 1)) + 1| + 3)) - 3) - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{... | -4 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = 2sin(3π((-2(-1(-2|x - 3| - 3)^2 + 3(-2|x - 3| - 3) + 2) + 1)/(-3(-1(-2|x - 3| - 3)^2 + 3(-2|x - 3| - 3) + 2) - 3)) - 2) + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 3 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = 1cos(3π((-3(-5(1|x + 2| - 1)^2 - 4(1|x + 2| - 1) + 1) + 1)/(1(-5(1|x + 2| - 1)^2 - 4(1|x + 2| - 1) + 1) - 3)) + 1) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex for... | -4 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = 5(-1|((2(-2sin(1πx - 2) + 2) + 1)/(2(-2sin(1πx - 2) + 2))) + 2| - 2)^2 + 4 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the largest y-value of the function f(x) = -1|(-1(-2cos(1π((3x - 2)/(1x + 2)) - 2) + 3)^2 - 1(-2cos(1π((3x - 2)/(1x + 2)) - 2) + 3) - 4) + 3| + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed... | -7 | MATH |
[
{
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = -2((1(3sin(2π(3|x - 3| + 3)) - 3) + 2)/(-2(3sin(2π(3|x - 3| + 3)) - 3) + 3))^2 - 4((1(3sin(2π(3|x - 3| + 3)) - 3) + 2)/(-2(3sin(2π(3|x - 3| + 3)) - 3) + 3)) - 5 in the interval [-10, 10]. Round to the nearest integer.\n\... | 0 | MATH |
[
{
"content": "How many local maxima does the function f(x) = -1|(-4(2cos(3π((1x + 3)/(1x - 2)) + 2))^2 - 1(2cos(3π((1x + 3)/(1x - 2)) + 2)) + 2) - 2| - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = (2(-1sin(1π(-3|(-1x^2 + 2x - 5) + 3| - 1) - 1) - 2))/(-2(-1sin(1π(-3|(-1x^2 + 2x - 5) + 3| - 1) - 1) - 2) - 2) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = (-1(5(1sin(2π(-2|x - 3| + 3) + 1) + 1)^2 + 1(1sin(2π(-2|x - 3| + 3) + 1) + 1)) - 1)/(3(5(1sin(2π(-2|x - 3| + 3) + 1) + 1)^2 + 1(1sin(2π(-2|x - 3| + 3) + 1) + 1))) in the interval [-10, 10]. Round to the nearest integer.\n\... | -10 | MATH |
[
{
"content": "Find the average of all x-coordinates of local maxima of the function f(x) = -1sin(1π(1|((1(1x^2 + 5x + 4) - 1)/(2(1x^2 + 5x + 4) + 2)) - 3| + 1)) - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -3 | MATH |
[
{
"content": "Find the average of all x-coordinates of local minima of the function f(x) = -1cos(1π(-1|(-4((3x - 2)/(-3x))^2 - 2) + 1| + 2) + 2) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "How many local maxima does the function f(x) = 2|(4((2(-1sin(3πx + 2) + 2) + 1)/(-1(-1sin(3πx + 2) + 2) + 2))^2 - 3) - 3| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local maximum of the function f(x) = 2sin(1π(-2|((1(-1x^2 - 3x - 1) + 1)/(1(-1x^2 - 3x - 1))) + 3| + 2)) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -4 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = 1|((-1(1(3sin(2πx + 2) - 3)^2 - 4(3sin(2πx + 2) - 3) + 3) - 1)/(3(1(3sin(2πx + 2) - 3)^2 - 4(3sin(2πx + 2) - 3) + 3) - 2)) - 2| + 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the ans... | -10 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = 2|(-1sin(2π(-5((-1x - 2)/(2x - 1))^2 - 1((-1x - 2)/(2x - 1)) + 4))) + 2| + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | -1 | MATH |
[
{
"content": "Find the x-coordinate of the leftmost local minimum of the function f(x) = 2cos(2π((1(2(3|x - 2| - 1)^2 + 4(3|x - 2| - 1)))/(3(2(3|x - 2| - 1)^2 + 4(3|x - 2| - 1)) + 1)) - 2) - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 2 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the smallest y-value of the function f(x) = -1((3(-2|(-2cos(1πx) - 1) + 3|) + 3)/(-1(-2|(-2cos(1πx) - 1) + 3|) + 3))^2 - 4((3(-2|(-2cos(1πx) - 1) + 3|) + 3)/(-1(-2|(-2cos(1πx) - 1) + 3|) + 3)) + 3 in the interval [-10, 10]. Round to the nearest integ... | -8 | MATH |
[
{
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = -1cos(3π((-3(4(-2|x|)^2 + 2(-2|x|) - 4) - 1)/(3(4(-2|x|)^2 + 2(-2|x|) - 4)))) + 1 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | -1 | MATH |
[
{
"content": "Find the x-coordinate of the local minimum with the largest y-value of the function f(x) = -1cos(1π(-3(-2|((3x + 1)/(2x + 2)) + 3| + 2)^2 + 1(-2|((3x + 1)/(2x + 2)) + 3| + 2) - 4) - 3) + 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your ... | -1 | MATH |
[
{
"content": "How many local extrema (both minima and maxima) does the function f(x) = 3|(3sin(3π((3(-5x^2 - 4x - 5) + 2)/(-2(-5x^2 - 4x - 5) + 3)) + 1) + 1) + 3| - 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = -3(-2sin(3π(1|((3x + 3)/(-3x + 1)) + 2|) - 2) - 1)^2 + 1(-2sin(3π(1|((3x + 3)/(-3x + 1)) + 2|) - 2) - 1) - 4 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\... | -4 | MATH |
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