messages listlengths 1 1 | ground_truth stringclasses 80 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "For the function f(x) = -2(4(-1|(-5(3sin(1π((2x - 1)/(3x + 3)) - 2) + 3)^2 + 1) + 3| - 2)) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(3sin(1π(1|((2(2|((-1x - 1)/(1x - 2)) + 1| - 2) + 2)/(3(2|((-1x - 1)/(1x - 2)) + 1| - 2) - 1))| - 1) - 3) + 2) - 3)/(-3(3sin(1π(1|((2(2|((-1x - 1)/(1x - 2)) + 1| - 2) + 2)/(3(2|((-1x - 1)/(1x - 2)) + 1| - ... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(3|(1(-4((2(3cos(1πx - 2) - 3) - 2)/(1(3cos(1πx - 2) - 3))) - 1) + 5) - 1| - 1) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((1(5((3(3(2cos(2πx + 2) - 2) + 2) + 3)/(2(3(2cos(2πx + 2) - 2) + 2) - 1))^2 - 4((3(3(2cos(2πx + 2) - 2) + 2) + 3)/(2(3(2cos(2πx + 2) - 2) + 2) - 1)) - 1) + 2)/(2(5((3(3(2cos(2πx + 2) - 2) + 2) + 3)/(2(3(2cos(2π... | 44 | MATH |
[
{
"content": "For the function f(x) = 3(5((2(-1|(3(3sin(2πx + 1) + 3)^2 + 1(3sin(2πx + 1) + 3) + 3) - 2| + 3) - 1)/(3(-1|(3(3sin(2πx + 1) + 3)^2 + 1(3sin(2πx + 1) + 3) + 3) - 2| + 3) - 2)) - 1)^2 + 2(5((2(-1|(3(3sin(2πx + 1) + 3)^2 + 1(3sin(2πx + 1) + 3) + 3) - 2| + 3) - 1)/(3(-1|(3(3sin(2πx + 1) + 3)^2 + 1(3si... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(2|(-1(1(-1(-1sin(1πx) + 3)^2 + 2(-1sin(1πx) + 3) - 2)) + 4) + 1| + 1) - 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(2|(-3cos(1π(-3(3(-1x^2 - 5x - 5) - 1)^2 + 5(3(-1x^2 - 5x - 5) - 1) + 2) + 1) - 1)| - 1) + 3| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = (-3(-1((1(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) - 1) + 2)/(3(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) - 1) + 1)) + 2))/(2(-1((1(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) - 1) + 2)/(3(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) - 1) + 1)) + 2) - 1), find the number of maximal... | 5 | MATH |
[
{
"content": "For the function f(x) = -3(-1(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5)^2 + 4(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5) - 5)^2 + 2(-1(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5)^2 + 4... | 18 | MATH |
[
{
"content": "For the function f(x) = 3|(-1|(3((1(-1(3sin(1πx - 1) + 2) + 4) + 2)/(-1(-1(3sin(1πx - 1) + 2) + 4))) + 2) + 1| + 1) - 1|, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|((3(2cos(3π(1(5((-1x + 1)/(2x)) - 4)) + 3)) + 1)/(-3(2cos(3π(1(5((-1x + 1)/(2x)) - 4)) + 3)) + 1)) + 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "For the function f(x) = 2|(4(2((2(2(1cos(3πx - 3) + 3) - 1) + 1)/(-2(2(1cos(3πx - 3) + 3) - 1) - 3))) - 1)| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((2(-2(2(-3cos(3π((-2x + 3)/(2x - 1)) - 1) - 2) + 2)^2 - 4) - 1)/(1(-2(2(-3cos(3π((-2x + 3)/(2x - 1)) - 1) - 2) + 2)^2 - 4) + 2))^2 + 1((2(-2(2(-3cos(3π((-2x + 3)/(2x - 1)) - 1) - 2) + 2)^2 - 4) - 1)/(1(-2(2(-3c... | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((-2(3sin(3π((1(1(-1x^2 + 2x + 5) + 2))/(1(1(-1x^2 + 2x + 5) + 2) - 2)) - 3) - 1) - 1)/(2(3sin(3π((1(1(-1x^2 + 2x + 5) + 2))/(1(1(-1x^2 + 2x + 5) + 2) - 2)) - 3) - 1) + 2))| + 1 negative?\n\nPresent the answer i... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-3(3|(-1|(1cos(3π((-3x + 1)/(2x - 3)) - 2) + 3) + 1| + 2) - 2| + 3)^2 - 2(3|(-1|(1cos(3π((-3x + 1)/(2x - 3)) - 2) + 3) + 1| + 2) - 2| + 3) - 4) + 1| - 2 is strictly increasing.\n\nPresent the answer in L... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(-3((2((-2(-1x^2 + 3x + 5) - 3)/(-2(-1x^2 + 3x + 5) + 3)) + 1)/(1((-2(-1x^2 + 3x + 5) - 3)/(-2(-1x^2 + 3x + 5) + 3)) + 1))^2) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Y... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3(1(-1cos(2π(1|(5x^2 + 5x + 1) + 1| - 3) - 2) - 2)^2 - 2(-1cos(2π(1|(5x^2 + 5x + 1) + 1| - 3) - 2) - 2) - 2)^2 - 3(1(-1cos(2π(1|(5x^2 + 5x + 1) + 1| - 3) - 2) - 2)^2 - 2(-1cos(2π(1|(5x^2 + 5x + 1) + 1| ... | 5 | MATH |
[
{
"content": "For the function f(x) = -1(2|(-1(3|(1(1cos(2πx + 1) + 1)^2 - 3(1cos(2πx + 1) + 1) - 2) - 1| - 2) - 5)| + 3) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|((-1(-2|(1(-1sin(1π(-2x + 2)) - 3)^2 + 3(-1sin(1π(-2x + 2)) - 3))| - 3) - 2)/(-3(-2|(1(-1sin(1π(-2x + 2)) - 3)^2 + 3(-1sin(1π(-2x + 2)) - 3))| - 3) - 1)) - 3| + 2 negative?\n\nPresent the answer in LaTex format... | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-2(-3|(4(-2(-2sin(1πx - 3) - 1)^2 - 3(-2sin(1πx - 3) - 1) + 3)^2 + 3(-2(-2sin(1πx - 3) - 1)^2 - 3(-2sin(1πx - 3) - 1) + 3) + 1) - 3| - 1) - 1) + 3| - 2 is strictly decreasing.\n\nPresent the answer in La... | 7 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2cos(1π(4(5((3(-5x) - 3)/(-1(-5x) - 2))^2 + 5((3(-5x) - 3)/(-1(-5x) - 2)) - 4)) - 1) + 1) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 41 | MATH |
[
{
"content": "For the function f(x) = (1(-2|(1(4(2|(-3x^2 + 2x + 1) + 2| - 1)^2 - 3(2|(-3x^2 + 2x + 1) + 2| - 1)) + 1) - 3| - 1) + 1)/(-3(-2|(1(4(2|(-3x^2 + 2x + 1) + 2| - 1)^2 - 3(2|(-3x^2 + 2x + 1) + 2| - 1)) + 1) - 3| - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\n... | 4 | MATH |
[
{
"content": "For the function f(x) = 3((-2(4(-2cos(1π(3|(-1x + 1) - 1|) + 3) - 1)) + 3)/(-2(4(-2cos(1π(3|(-1x + 1) - 1|) + 3) - 1))))^2 + 2((-2(4(-2cos(1π(3|(-1x + 1) - 1|) + 3) - 1)) + 3)/(-2(4(-2cos(1π(3|(-1x + 1) - 1|) + 3) - 1)))) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x)... | 30 | MATH |
[
{
"content": "For the function f(x) = -3((-3(-1(3sin(3π(-1|((1x - 1)/(-3x - 3))| + 1) + 2)) + 3) + 3)/(-3(-1(3sin(3π(-1|((1x - 1)/(-3x - 3))| + 1) + 2)) + 3)))^2 - 1((-3(-1(3sin(3π(-1|((1x - 1)/(-3x - 3))| + 1) + 2)) + 3) + 3)/(-3(-1(3sin(3π(-1|((1x - 1)/(-3x - 3))| + 1) + 2)) + 3))), find the number of maximal... | 10 | MATH |
[
{
"content": "For the function f(x) = 1sin(2π(3(4(4(1((-2x + 1)/(1x - 2))^2 - 3((-2x + 1)/(1x - 2)) + 1) - 1)) + 3) + 2) + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2((-1(-1|((1(-3(-3sin(2πx + 2) - 3) - 3) - 2)/(-3(-3(-3sin(2πx + 2) - 3) - 3) - 1))| + 3) - 1)/(-1(-1|((1(-3(-3sin(2πx + 2) - 3) - 3) - 2)/(-3(-3(-3sin(2πx + 2) - 3) - 3) - 1))| + 3) + 2)) + 1)/(-3((-1(-1|(... | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-2sin(2π((-1(-4((1(1x + 5) + 2)/(3(1x + 5) - 1))^2 + 5((1(1x + 5) + 2)/(3(1x + 5) - 1)) - 1) + 1)/(-2(-4((1(1x + 5) + 2)/(3(1x + 5) - 1))^2 + 5((1(1x + 5) + 2)/(3(1x + 5) - 1)) - 1) - 2)) - 2)) is strictly... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3cos(3π(3(-5((-2(1x + 2) + 1)/(-2(1x + 2) + 3)) - 3)^2 - 1(-5((-2(1x + 2) + 1)/(-2(1x + 2) + 3)) - 3) - 2) + 1) - 1) + 1)/(3(3cos(3π(3(-5((-2(1x + 2) + 1)/(-2(1x + 2) + 3)) - 3)^2 - 1(-5((-2(1x + 2) + 1)/(-2(1x... | 50 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2((1((-2(-1cos(2π(-3|(-1x^2 - 3x + 2) - 1|) - 2) - 1) - 1)/(2(-1cos(2π(-3|(-1x^2 - 3x + 2) - 1|) - 2) - 1) - 3)) - 3)/(1((-2(-1cos(2π(-3|(-1x^2 - 3x + 2) - 1|) - 2) - 1) - 1)/(2(-1cos(2π(-3|(-1x^2 - 3x + 2)... | 50 | MATH |
[
{
"content": "For the function f(x) = 1|(-2(-2|(-4(-3sin(1π(3|x + 1| - 1) - 2) + 2) + 2) - 2| + 1)^2 + 1(-2|(-4(-3sin(1π(3|x + 1| - 1) - 2) + 2) + 2) - 2| + 1) - 2) + 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1(-2|((3(2sin(3π(2x + 3) - 1) - 1) + 3)/(2(2sin(3π(2x + 3) - 1) - 1))) - 2| + 3) - 3) + 1)/(-1(-1(-2|((3(2sin(3π(2x + 3) - 1) - 1) + 3)/(2(2sin(3π(2x + 3) - 1) - 1))) - 2| + 3) - 3) + 2) is strictly incr... | 63 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((-3(3(-3|(2(-2x^2 - 2x + 4)^2 - 1(-2x^2 - 2x + 4) - 5) - 2| - 1)^2 + 4(-3|(2(-2x^2 - 2x + 4)^2 - 1(-2x^2 - 2x + 4) - 5) - 2| - 1) + 5) - 2)/(3(3(-3|(2(-2x^2 - 2x + 4)^2 - 1(-2x^2 - 2x + 4) - 5) - 2| - 1)^... | 4 | MATH |
[
{
"content": "For the function f(x) = -2(-3|(-4(1(5((1x + 3)/(3x))^2 + 3((1x + 3)/(3x))) - 3)^2 + 3(1(5((1x + 3)/(3x))^2 + 3((1x + 3)/(3x))) - 3)) + 3|)^2 - 4(-3|(-4(1(5((1x + 3)/(3x))^2 + 3((1x + 3)/(3x))) - 3)^2 + 3(1(5((1x + 3)/(3x))^2 + 3((1x + 3)/(3x))) - 3)) + 3|), find the number of maximal connected int... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((3(-1|(3|(1cos(3π((-2x - 2)/(-1x - 3))) + 3) + 1|)| + 3) + 3)/(2(-1|(3|(1cos(3π((-2x - 2)/(-1x - 3))) + 3) + 1|)| + 3) + 1)) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your... | 15 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-2(-1|(1((1(2cos(3πx + 2)) + 3)/(-2(2cos(3πx + 2)) - 1))^2 - 1((1(2cos(3πx + 2)) + 3)/(-2(2cos(3πx + 2)) - 1)) + 1) + 3| + 1) + 4) + 2| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \... | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2π(2(3(1|((2(5x - 3) - 2)/(1(5x - 3) - 2)) + 2| - 1) + 1) - 2) + 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -1|(-3|(2|(2((3(-1cos(3πx - 3) + 3) + 3)/(2(-1cos(3πx - 3) + 3) - 3))^2 + 5((3(-1cos(3πx - 3) + 3) + 3)/(2(-1cos(3πx - 3) + 3) - 3)) + 5) + 1| + 3) - 3| + 1)| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex fo... | 30 | MATH |
[
{
"content": "For the function f(x) = (2(-4(2|((-2(-2sin(1π((-1x + 2)/(-1x - 3))) - 3) - 1)/(2(-2sin(1π((-1x + 2)/(-1x - 3))) - 3) + 2)) - 2| - 2) - 5) - 2)/(2(-4(2|((-2(-2sin(1π((-1x + 2)/(-1x - 3))) - 3) - 1)/(2(-2sin(1π((-1x + 2)/(-1x - 3))) - 3) + 2)) - 2| - 2) - 5) - 3), find the number of maximal connecte... | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|((2(1(-3cos(1π(-3(-3|x - 3| + 3)^2 - 4(-3|x - 3| + 3) - 5) - 2) - 2)^2 + 5) + 2)/(-3(1(-3cos(1π(-3(-3|x - 3| + 3)^2 - 4(-3|x - 3| + 3) - 5) - 2) - 2)^2 + 5) + 1)) + 1| + 3 positive?\n\nPresent the answer in LaTe... | 50 | MATH |
[
{
"content": "For the function f(x) = 4(2cos(3π(4((3(-2(-1x - 3)^2 + 4(-1x - 3)) + 2)/(2(-2(-1x - 3)^2 + 4(-1x - 3)) - 1)) - 5) - 1) + 1)^2 - 3(2cos(3π(4((3(-2(-1x - 3)^2 + 4(-1x - 3)) + 2)/(2(-2(-1x - 3)^2 + 4(-1x - 3)) - 1)) - 5) - 1) + 1) + 4, find the number of maximal connected intervals in [-10, 10] where... | 16 | MATH |
[
{
"content": "For the function f(x) = -4(-1(-1(-3cos(2π(2((-1x + 2)/(-2x - 3))^2 - 1((-1x + 2)/(-2x - 3))) + 3) + 3) - 1)^2 + 2(-1(-3cos(2π(2((-1x + 2)/(-2x - 3))^2 - 1((-1x + 2)/(-2x - 3))) + 3) + 3) - 1) + 5) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answ... | 5 | MATH |
[
{
"content": "For the function f(x) = -1cos(3π(3|(-1(-1(-3(2|x - 3| + 1) + 5) - 2)^2 - 1(-1(-3(2|x - 3| + 1) + 5) - 2) + 1) + 1| - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2cos(2π(-2|((3(4((-2x + 3)/(3x - 2))^2 - 3((-2x + 3)/(3x - 2)) + 2) + 3)/(3(4((-2x + 3)/(3x - 2))^2 - 3((-2x + 3)/(3x - 2)) + 2) - 2)) - 1|))) - 3)/(-3(2cos(2π(-2|((3(4((-2x + 3)/(3x - 2))^2 - 3((-2x + 3)... | 7 | MATH |
[
{
"content": "For the function f(x) = 1((1(-5(1|(1(5x^2 - 2x - 2)^2 - 3(5x^2 - 2x - 2) - 4) + 2| - 3)^2 - 1(1|(1(5x^2 - 2x - 2)^2 - 3(5x^2 - 2x - 2) - 4) + 2| - 3) - 1))/(2(-5(1|(1(5x^2 - 2x - 2)^2 - 3(5x^2 - 2x - 2) - 4) + 2| - 3)^2 - 1(1|(1(5x^2 - 2x - 2)^2 - 3(5x^2 - 2x - 2) - 4) + 2| - 3) - 1) - 1))^2 - 2, ... | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3|(-5((2(-3sin(3π(1x^2 - 1x - 3)) - 2))/(-1(-3sin(3π(1x^2 - 1x - 3)) - 2) - 2))^2 - 5((2(-3sin(3π(1x^2 - 1x - 3)) - 2))/(-1(-3sin(3π(1x^2 - 1x - 3)) - 2) - 2)) + 5) + 3| + 3) + 5 negative?\n\nPresent the answer... | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(3|((2(2(-2cos(1π(-1|x - 3| - 1)))^2 - 1(-2cos(1π(-1|x - 3| - 1))) + 1) + 3)/(2(2(-2cos(1π(-1|x - 3| - 1)))^2 - 1(-2cos(1π(-1|x - 3| - 1))) + 1) + 2)) + 2| - 3) - 2| + 2 positive?\n\nPresent the answer in LaTex... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(1(2(3cos(3π(-3|(3|x + 1| + 1) - 3| + 2) - 2)) - 4) + 2) - 3)/(-1(1(2(3cos(3π(-3|(3|x + 1| + 1) - 3| + 2) - 2)) - 4) + 2) + 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 58 | MATH |
[
{
"content": "For the function f(x) = 2|(-1cos(1π(2(4(-4(1x - 2)^2 + 2(1x - 2) + 2)^2 + 2(-4(1x - 2)^2 + 2(1x - 2) + 2) - 4) + 2)) + 2) + 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((1(-1|(-4(3|(-3cos(2πx - 3) + 2) - 2| + 1) + 2)| - 3) + 2)/(-3(-1|(-4(3|(-3cos(2πx - 3) + 2) - 2| + 1) + 2)| - 3) + 1)) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 40 | MATH |
[
{
"content": "For the function f(x) = -1(1cos(3π(-3(-4((1((-3x + 3)/(2x - 1)) - 2)/(-2((-3x + 3)/(2x - 1)) + 1)) + 3) + 1)) - 3)^2 - 2(1cos(3π(-3(-4((1((-3x + 3)/(2x - 1)) - 2)/(-2((-3x + 3)/(2x - 1)) + 1)) + 3) + 1)) - 3) + 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPres... | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(1|(1|(-1|(3sin(3π(1x - 2) + 2) + 3) + 2|) - 1| - 3) - 1| - 3)^2 - 2(1|(1|(-1|(3sin(3π(1x - 2) + 2) + 3) + 2|) - 1| - 3) - 1| - 3) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{... | 83 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2|(-2|(-1(2(-1cos(1πx - 2) + 3)^2 - 2(-1cos(1πx - 2) + 3) - 4)) + 3| + 2) - 1| + 3)| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "For the function f(x) = -4(2(-3(1cos(3π(-3(1x + 4)^2 + 4(1x + 4) - 3) + 1) - 1)^2 + 5(1cos(3π(-3(1x + 4)^2 + 4(1x + 4) - 3) + 1) - 1))^2 + 4(-3(1cos(3π(-3(1x + 4)^2 + 4(1x + 4) - 3) + 1) - 1)^2 + 5(1cos(3π(-3(1x + 4)^2 + 4(1x + 4) - 3) + 1) - 1)) + 4) - 2, find the number of maximal connected inte... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-2|(2cos(1π(-4(5(-1x^2 - 3x + 1) - 5) - 3) - 3) - 1) + 1| - 3) - 3)/(2(-2|(2cos(1π(-4(5(-1x^2 - 3x + 1) - 5) - 3) - 3) - 1) + 1| - 3)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxe... | 63 | MATH |
[
{
"content": "For the function f(x) = (-2(-5(2(-2|(3sin(3π(1|x + 1|) + 3) - 2) - 2| + 1))^2 - 5(2(-2|(3sin(3π(1|x + 1|) + 3) - 2) - 2| + 1)) + 5))/(1(-5(2(-2|(3sin(3π(1|x + 1|) + 3) - 2) - 2| + 1))^2 - 5(2(-2|(3sin(3π(1|x + 1|) + 3) - 2) - 2| + 1)) + 5) - 3), find the number of maximal connected intervals in [-... | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1(1sin(1π(-3(-1((1x + 3)/(-3x - 3)) + 4)^2 - 4(-1((1x + 3)/(-3x - 3)) + 4) + 5) - 2) + 2)^2 + 2(1sin(1π(-3(-1((1x + 3)/(-3x - 3)) + 4)^2 - 4(-1((1x + 3)/(-3x - 3)) + 4) + 5) - 2) + 2) + 5) - 2)/(1(-1(1sin(1π(... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-1(-3|(-3sin(1π((1((1x)/(-1x + 2)))/(-2((1x)/(-1x + 2)) - 1)) - 3) + 1) + 2| - 3) - 1) + 2)/(-2(-1(-3|(-3sin(1π((1((1x)/(-1x + 2)))/(-2((1x)/(-1x + 2)) - 1)) - 3) + 1) + 2| - 3) - 1) - 2) positive?\n\nPresent ... | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-1(-2|(-1|((1(-2cos(3πx - 3) - 1) - 3)/(1(-2cos(3πx - 3) - 1) - 2)) + 3| - 1) + 2| + 3) - 3)/(3(-2|(-1|((1(-2cos(3πx - 3) - 1) - 3)/(1(-2cos(3πx - 3) - 1) - 2)) + 3| - 1) + 2| + 3) + 3))^2 + 3((-1(-2|(-1|((1(-... | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(2|(-2(1(-1(1cos(1πx - 2) - 1)^2 - 4(1cos(1πx - 2) - 1) + 4) + 4) - 4) + 1| + 3)^2 - 3(2|(-2(1(-1(1cos(1πx - 2) - 1)^2 - 4(1cos(1πx - 2) - 1) + 4) + 4) - 4) + 1| + 3) + 2 is strictly increasing.\n\nPresent ... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(1(-4(4(2cos(2π(-3x^2 - 5x + 5)) + 3)^2 + 4(2cos(2π(-3x^2 - 5x + 5)) + 3) + 3) - 2) - 1))/(-2(1(-4(4(2cos(2π(-3x^2 - 5x + 5)) + 3)^2 + 4(2cos(2π(-3x^2 - 5x + 5)) + 3) + 3) - 2) - 1) - 2) negative?\n\nPresent the... | 50 | MATH |
[
{
"content": "For the function f(x) = 3(-5(1|(-2(1cos(3π(-1|x| + 3) - 3) - 2)^2 + 3(1cos(3π(-1|x| + 3) - 3) - 2) + 2) - 2| - 3) - 2) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-4(1cos(3π(-5(2|(1x^2 + 3x - 2) + 2| + 1) - 4) + 2) + 1) + 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5((1((2(3(-2|(-2sin(3πx - 2)) - 2|)) + 2)/(-3(3(-2|(-2sin(3πx - 2)) - 2|)) + 4)) - 1)/(1((2(3(-2|(-2sin(3πx - 2)) - 2|)) + 2)/(-3(3(-2|(-2sin(3πx - 2)) - 2|)) + 4)) + 2))^2 + 3 positive?\n\nPresent the answer in L... | 30 | MATH |
[
{
"content": "For the function f(x) = -2|(-5(-4((1(-3(1x^2 + 5x + 3) + 4))/(-2(-3(1x^2 + 5x + 3) + 4) + 1))^2 - 1((1(-3(1x^2 + 5x + 3) + 4))/(-2(-3(1x^2 + 5x + 3) + 4) + 1))) - 1)| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((-1(1|(-2|(-2(-1|x - 3| - 3) - 4) - 3|) + 2| - 2))/(2(1|(-2|(-2(-1|x - 3| - 3) - 4) - 3|) + 2| - 2) + 3))^2 - 1((-1(1|(-2|(-2(-1|x - 3| - 3) - 4) - 3|) + 2| - 2))/(2(1|(-2|(-2(-1|x - 3| - 3) - 4) - 3|) + 2| - 2)... | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(1sin(3π(-5(-3|((2(-5x^2 - 1x - 5) + 3)/(1(-5x^2 - 1x - 5) + 3)) + 1|) - 4) + 2) - 1) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3(1|(-1|(-1(3cos(2πx + 2) - 3)^2 - 3(3cos(2πx + 2) - 3) - 3) + 3| - 3) - 1| + 1) - 2)/(1(1|(-1|(-1(3cos(2πx + 2) - 3)^2 - 3(3cos(2πx + 2) - 3) - 3) + 3| - 3) - 1| + 1)))^2 - 1 is strictly increasing.\n\n... | 55 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(2|(-2(1(-3(-2cos(3πx - 2) + 1) + 2)^2 - 4(-3(-2cos(3πx - 2) + 1) + 2)) + 4) - 1| + 3) + 1)/(2(2|(-2(1(-3(-2cos(3πx - 2) + 1) + 2)^2 - 4(-3(-2cos(3πx - 2) + 1) + 2)) + 4) - 1| + 3) - 1) is strictly increa... | 33 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1((2((-1(-1|((-3x + 2)/(1x + 3)) + 3| + 3) + 3)/(-3(-1|((-3x + 2)/(1x + 3)) + 3| + 3) - 2)) + 2)/(-1((-1(-1|((-3x + 2)/(1x + 3)) + 3| + 3) + 3)/(-3(-1|((-3x + 2)/(1x + 3)) + 3| + 3) - 2)) + 2))^2 + 1((2((-1(-1... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3cos(3π(4(-5(1(4x^2 + 1)^2 - 1(4x^2 + 1) - 3)^2 + 4(1(4x^2 + 1)^2 - 1(4x^2 + 1) - 3) - 1) - 1)) + 1) + 1)/(-3(-3cos(3π(4(-5(1(4x^2 + 1)^2 - 1(4x^2 + 1) - 3)^2 + 4(1(4x^2 + 1)^2 - 1(4x^2 + 1) - 3) - 1) -... | 16 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-3(2cos(2π(1(3|((-2x + 1)/(3x - 2))| + 3)^2 + 1(3|((-2x + 1)/(3x - 2))| + 3) - 2) + 2) + 2) - 2) - 2| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "For the function f(x) = (1(4(-1|(4(-4(-1sin(2πx - 1) + 3)^2 - 1(-1sin(2πx - 1) + 3) + 5) + 1) - 2| + 2) - 2) - 1)/(2(4(-1|(4(-4(-1sin(2πx - 1) + 3)^2 - 1(-1sin(2πx - 1) + 3) + 5) + 1) - 2| + 2) - 2) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the a... | 21 | MATH |
[
{
"content": "For the function f(x) = -2(1|(3sin(1π(-3(2(-1x - 4) + 4)^2 + 4(2(-1x - 4) + 4) - 4) - 3)) - 3| + 1)^2 - 4(1|(3sin(1π(-3(2(-1x - 4) + 4)^2 + 4(2(-1x - 4) + 4) - 4) - 3)) - 3| + 1) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2|(-3cos(2π(4(-2|(5x - 1) - 2|) + 4)) + 3) - 3| + 2) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1cos(3π(3(1|(3|(1|x - 2| + 2) + 1|) - 2| + 2)^2 + 3(1|(3|(1|x - 2| + 2) + 1|) - 2| + 2) + 2) - 2) + 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 3(-2sin(1π(-3(-5((-2(1x + 3) + 1)/(-1(1x + 3) - 3))^2 + 1((-2(1x + 3) + 1)/(-1(1x + 3) - 3)) + 2)^2 + 2(-5((-2(1x + 3) + 1)/(-1(1x + 3) - 3))^2 + 1((-2(1x + 3) + 1)/(-1(1x + 3) - 3)) + 2)) + 2) + 1)^2 + 5, find the number of maximal connected intervals in [-10, 10] where f'... | 21 | MATH |
[
{
"content": "For the function f(x) = (-3(-1(3(2((3(4x + 1) - 1)/(2(4x + 1) + 2))^2 - 4((3(4x + 1) - 1)/(2(4x + 1) + 2)))^2 + 4(2((3(4x + 1) - 1)/(2(4x + 1) + 2))^2 - 4((3(4x + 1) - 1)/(2(4x + 1) + 2))) - 5)^2 - 3(3(2((3(4x + 1) - 1)/(2(4x + 1) + 2))^2 - 4((3(4x + 1) - 1)/(2(4x + 1) + 2)))^2 + 4(2((3(4x + 1) - ... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1((1(5((2(-3sin(2πx - 1) + 1) - 3)/(1(-3sin(2πx - 1) + 1) - 2))^2 - 3((2(-3sin(2πx - 1) + 1) - 3)/(1(-3sin(2πx - 1) + 1) - 2)) - 4) + 3)/(-2(5((2(-3sin(2πx - 1) + 1) - 3)/(1(-3sin(2πx - 1) + 1) - 2))^2 - 3((2(-... | 37 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5(-3sin(1π((-3(2|((-2(3|x| - 3) - 3)/(-3(3|x| - 3) + 4)) - 1| + 1))/(-3(2|((-2(3|x| - 3) - 3)/(-3(3|x| - 3) + 4)) - 1| + 1) - 1)) - 1) + 2)^2 - 4(-3sin(1π((-3(2|((-2(3|x| - 3) - 3)/(-3(3|x| - 3) + 4)) - 1| ... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3(3sin(3π(-1(4(3|x - 2| - 1)) + 3) + 1) - 2) + 4) - 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = (1((1(2sin(2π((1(-3((3x - 1)/(-3x - 1)) - 5) - 1)/(-3(-3((3x - 1)/(-3x - 1)) - 5))) + 1) + 1) - 3)/(2(2sin(2π((1(-3((3x - 1)/(-3x - 1)) - 5) - 1)/(-3(-3((3x - 1)/(-3x - 1)) - 5))) + 1) + 1))) - 3)/(-1((1(2sin(2π((1(-3((3x - 1)/(-3x - 1)) - 5) - 1)/(-3(-3((3x - 1)/(-3x - 1))... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(1π((-1((-2(5((-1(-3x + 4) - 1)/(-1(-3x + 4)))^2 - 4((-1(-3x + 4) - 1)/(-1(-3x + 4))) + 3) - 1)/(-1(5((-1(-3x + 4) - 1)/(-1(-3x + 4)))^2 - 4((-1(-3x + 4) - 1)/(-1(-3x + 4))) + 3) - 3)) + 1)/(-1((-2(5((-1... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(1|(4(2|(2(-2sin(3πx + 3) + 2) - 1) - 1| + 3) + 1) - 2| + 2) + 1)/(-2(1|(4(2|(2(-2sin(3πx + 3) + 2) - 1) - 1| + 3) + 1) - 2| + 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 60 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1|(-2|(-1(3|(3sin(1πx + 2) - 2)| - 1) + 5) + 1| + 2) + 2|)^2 - 4(1|(-2|(-1(3|(3sin(1πx + 2) - 2)| - 1) + 5) + 1| + 2) + 2|) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 32 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((-3((1(3|(-2cos(1π(-2x^2 - 3x - 1) - 1) + 3) + 1| + 2) - 2)/(2(3|(-2cos(1π(-2x^2 - 3x - 1) - 1) + 3) + 1| + 2) + 3)) + 2)/(-2((1(3|(-2cos(1π(-2x^2 - 3x - 1) - 1) + 3) + 1| + 2) - 2)/(2(3|(-2cos(1π(-2x^2 - 3x - ... | 46 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1|((-2(1(3(3sin(3πx - 3))^2 - 4)^2 + 2(3(3sin(3πx - 3))^2 - 4) + 5) - 1)/(3(1(3(3sin(3πx - 3))^2 - 4)^2 + 2(3(3sin(3πx - 3))^2 - 4) + 5) - 2))| + 2) - 5 is strictly increasing.\n\nPresent the answer in La... | 60 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -5((-2((2(-3((-1(1cos(2πx + 1) - 1) - 1)/(2(1cos(2πx + 1) - 1)))^2 + 3((-1(1cos(2πx + 1) - 1) - 1)/(2(1cos(2πx + 1) - 1))) + 5) + 3)/(-2(-3((-1(1cos(2πx + 1) - 1) - 1)/(2(1cos(2πx + 1) - 1)))^2 + 3((-1(1cos(... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(3cos(3π(2((3(2((-3x - 2)/(2x)) + 1) + 2)/(2(2((-3x - 2)/(2x)) + 1) + 1))^2 - 1((3(2((-3x - 2)/(2x)) + 1) + 2)/(2(2((-3x - 2)/(2x)) + 1) + 1)) + 4) - 1) + 2) + 2)/(2(3cos(3π(2((3(2((-3x - 2)/(2x)) + 1) + 2)/(2(2... | 9 | MATH |
[
{
"content": "For the function f(x) = -2sin(1π(2(-2|(5((-1(-5x - 2) + 2)/(-2(-5x - 2))) + 3) + 3| - 1)^2 - 3(-2|(5((-1(-5x - 2) + 2)/(-2(-5x - 2))) + 3) + 3| - 1) + 2) - 2) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 38 | MATH |
[
{
"content": "For the function f(x) = 5((2(-2(-2cos(3π((-1((-1x + 2)/(3x)) + 2)/(2((-1x + 2)/(3x)) - 1))) + 3)^2 - 3(-2cos(3π((-1((-1x + 2)/(3x)) + 2)/(2((-1x + 2)/(3x)) - 1))) + 3) + 1))/(3(-2(-2cos(3π((-1((-1x + 2)/(3x)) + 2)/(2((-1x + 2)/(3x)) - 1))) + 3)^2 - 3(-2cos(3π((-1((-1x + 2)/(3x)) + 2)/(2((-1x + 2)/... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(5(-3sin(3π((2((-2(1x - 3))/(3(1x - 3) - 2)) + 2)/(-2((-2(1x - 3))/(3(1x - 3) - 2)) + 1)) + 2) + 3) + 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1π((-2(3(-4((-1(-3|x - 3| + 1))/(-1(-3|x - 3| + 1) - 3)) - 4)^2 - 3(-4((-1(-3|x - 3| + 1))/(-1(-3|x - 3| + 1) - 3)) - 4) - 1) - 2)/(-1(3(-4((-1(-3|x - 3| + 1))/(-1(-3|x - 3| + 1) - 3)) - 4)^2 - 3(-4((-1(-3|x... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1((-3(-2(4(-1|((-2x)/(-1x - 3))| + 2)^2 - 2(-1|((-2x)/(-1x - 3))| + 2) - 5)^2 + 5(4(-1|((-2x)/(-1x - 3))| + 2)^2 - 2(-1|((-2x)/(-1x - 3))| + 2) - 5) + 1) + 3)/(-1(-2(4(-1|((-2x)/(-1x - 3))| + 2)^2 - 2(-1|((... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(3cos(1π(2((3(3|(1x^2 - 5x + 4) - 2| - 1) - 1)/(-2(3|(1x^2 - 5x + 4) - 2| - 1) - 3))^2 - 2((3(3|(1x^2 - 5x + 4) - 2| - 1) - 1)/(-2(3|(1x^2 - 5x + 4) - 2| - 1) - 3))) + 3) - 1) - 3| - 2 is strictly increasi... | 12 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2π(-3|(-3(-1(-1((-1x - 2)/(2x - 1))^2 - 1((-1x - 2)/(2x - 1)) + 5)^2 - 2(-1((-1x - 2)/(2x - 1))^2 - 1((-1x - 2)/(2x - 1)) + 5) + 4)^2 + 3(-1(-1((-1x - 2)/(2x - 1))^2 - 1((-1x - 2)/(2x - 1)) + 5)^2 - 2(... | 25 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1(-4((1(-1cos(3π((-2x + 2)/(2x - 1)) + 1) + 3) + 3)/(2(-1cos(3π((-2x + 2)/(2x - 1)) + 1) + 3) - 1))^2 - 2)^2 + 1(-4((1(-1cos(3π((-2x + 2)/(2x - 1)) + 1) + 3) + 3)/(2(-1cos(3π((-2x + 2)/(2x - 1)) + 1) + 3) - 1))... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1π(4(1|((-3(2(-4x^2 + 4)^2 + 5(-4x^2 + 4) - 1) - 2)/(-3(2(-4x^2 + 4)^2 + 5(-4x^2 + 4) - 1) + 2)) + 2| - 3)^2 + 2(1|((-3(2(-4x^2 + 4)^2 + 5(-4x^2 + 4) - 1) - 2)/(-3(2(-4x^2 + 4)^2 + 5(-4x^2 + 4) - 1) + 2)) + ... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-2(2cos(3π(-3|(1((-1x + 1)/(-3x - 2))^2 + 1((-1x + 1)/(-3x - 2)) - 3) + 2| + 1) - 3)) - 3)^2 - 1(-2(2cos(3π(-3|(1((-1x + 1)/(-3x - 2))^2 + 1((-1x + 1)/(-3x - 2)) - 3) + 2| + 1) - 3)) - 3) + 1 negative?\n\nPrese... | 11 | MATH |
[
{
"content": "For the function f(x) = 2|(3(2|((-1((-1(-5x - 5) - 3)/(2(-5x - 5) - 1)) + 2)/(2((-1(-5x - 5) - 3)/(2(-5x - 5) - 1)))) + 3| - 3)^2 + 5(2|((-1((-1(-5x - 5) - 3)/(2(-5x - 5) - 1)) + 2)/(2((-1(-5x - 5) - 3)/(2(-5x - 5) - 1)))) + 3| - 3) - 2)| - 2, find the number of maximal connected intervals in [-10... | 4 | MATH |
[
{
"content": "For the function f(x) = -2sin(1π(2(-1(-5(-2|((-3x + 1)/(2x)) + 3| + 2) + 5) - 3)) + 2) - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
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