messages listlengths 1 1 | ground_truth stringclasses 76
values | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "For the function f(x) = -5(3(-1|(2sin(3π(-1x + 5) - 1) + 1) - 3| + 3)^2 + 3(-1|(2sin(3π(-1x + 5) - 1) + 1) - 3| + 3) - 5)^2 + 1(3(-1|(2sin(3π(-1x + 5) - 1) + 1) - 3| + 3)^2 + 3(-1|(2sin(3π(-1x + 5) - 1) + 1) - 3| + 3) - 5) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x... | 21 | MATH |
[
{
"content": "For the function f(x) = 3((1((1((1(-2sin(3πx + 3) + 2) + 2)/(1(-2sin(3πx + 3) + 2) + 3)) + 3)/(3((1(-2sin(3πx + 3) + 2) + 2)/(1(-2sin(3πx + 3) + 2) + 3)))) + 2)/(2((1((1(-2sin(3πx + 3) + 2) + 2)/(1(-2sin(3πx + 3) + 2) + 3)) + 3)/(3((1(-2sin(3πx + 3) + 2) + 2)/(1(-2sin(3πx + 3) + 2) + 3)))) - 3)) -... | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-1((3(3|(-1sin(1πx + 2) - 2)| - 1) - 1)/(2(3|(-1sin(1πx + 2) - 2)| - 1) - 1))^2 + 5((3(3|(-1sin(1πx + 2) - 2)| - 1) - 1)/(2(3|(-1sin(1πx + 2) - 2)| - 1) - 1)) + 4)^2 + 1(-1((3(3|(-1sin(1πx + 2) - 2)| - 1) - 1)/... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1(-1(-5(1sin(2πx) + 2)^2 - 2(1sin(2πx) + 2) - 3)^2 - 3(-5(1sin(2πx) + 2)^2 - 2(1sin(2πx) + 2) - 3) - 5) - 1) + 1)/(-2(-1(-1(-5(1sin(2πx) + 2)^2 - 2(1sin(2πx) + 2) - 3)^2 - 3(-5(1sin(2πx) + 2)^2 - 2(1sin(... | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(1(-2|(2cos(1π(1x + 2) - 1) + 3) + 1|)^2 - 5(-2|(2cos(1π(1x + 2) - 1) + 3) + 1|) + 5) - 5 is strictly increasing.\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) = 3|(-1|(-5(-2(3cos(3πx - 3) + 1)^2 + 2(3cos(3πx - 3) + 1) + 1) + 1) + 1| - 3) - 3| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "For the function f(x) = 5(-2((1(-2sin(3π(-5x + 1) - 1) - 1) - 3)/(-3(-2sin(3π(-5x + 1) - 1) - 1) - 2)) + 3)^2 - 3(-2((1(-2sin(3π(-5x + 1) - 1) - 1) - 3)/(-3(-2sin(3π(-5x + 1) - 1) - 1) - 2)) + 3) + 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answe... | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(2cos(2π(4(-2(-2|x - 1|)) + 5) - 3) - 3) + 3)/(-1(2cos(2π(4(-2(-2|x - 1|)) + 5) - 3) - 3) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 77 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2π(-3(1|((1((2x - 1)/(1x)) + 3)/(3((2x - 1)/(1x)) - 1)) + 1|)) + 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1π(2|((-1((2(1|x + 2| + 2) - 3)/(-2(1|x + 2| + 2))) + 2)/(1((2(1|x + 2| + 2) - 3)/(-2(1|x + 2| + 2))))) + 3| + 3) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3π(5(2(2(-1x^2 - 5x - 4) + 5) + 2) - 1) - 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "For the function f(x) = (1(-1(5(2sin(3π(-4x - 1) + 3) - 1)^2 + 4(2sin(3π(-4x - 1) + 3) - 1) - 3)) - 2)/(2(-1(5(2sin(3π(-4x - 1) + 3) - 1)^2 + 4(2sin(3π(-4x - 1) + 3) - 1) - 3))), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\b... | 42 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(1((-3(-1sin(2π((-2x + 3)/(3x)) + 1) - 2) - 1)/(-1(-1sin(2π((-2x + 3)/(3x)) + 1) - 2) + 2))^2 + 5) + 2| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-4(3(-3|(-3cos(2πx - 2) + 3) - 2| + 3) - 4)))/(3(-4(3(-3|(-3cos(2πx - 2) + 3) - 2| + 3) - 4)) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1(3(1cos(3π((-3x - 2)/(-2x + 3)) - 2)) + 3)^2 - 4(3(1cos(3π((-3x - 2)/(-2x + 3)) - 2)) + 3) - 1) + 1| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "For the function f(x) = (3(-5(5(4(-1sin(2πx - 3) + 1) + 5) - 3)^2 - 2(5(4(-1sin(2πx - 3) + 1) + 5) - 3) + 3) + 3)/(1(-5(5(4(-1sin(2πx - 3) + 1) + 5) - 3)^2 - 2(5(4(-1sin(2πx - 3) + 1) + 5) - 3) + 3) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the a... | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(3sin(2π((-2((2(-3|x + 3| + 3) - 2)/(-2(-3|x + 3| + 3) - 1)) - 1)/(-1((2(-3|x + 3| + 3) - 2)/(-2(-3|x + 3| + 3) - 1)) - 1)) + 2) - 2) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\box... | 41 | MATH |
[
{
"content": "For the function f(x) = 3|(4(-1sin(1π(-1|(1|x| + 2) + 1| + 1) + 3) + 1) - 2) - 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": "For the function f(x) = 3((3(1(-2|(-2sin(3πx - 3) + 3) + 2| + 1) + 5) + 1)/(-1(1(-2|(-2sin(3πx - 3) + 3) + 2| + 1) + 5))) + 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) = -2sin(3π(5(-1|(-1(3x - 2)^2 + 5(3x - 2)) + 1| + 2) - 1) + 1) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(4(-3(3|(-1cos(3πx - 3) + 1) + 3| - 2))^2 + 5(-3(3|(-1cos(3πx - 3) + 1) + 3| - 2)) + 4) + 1)/(3(4(-3(3|(-1cos(3πx - 3) + 1) + 3| - 2))^2 + 5(-3(3|(-1cos(3πx - 3) + 1) + 3| - 2)) + 4)) negative?\n\nPresent the an... | 30 | MATH |
[
{
"content": "For the function f(x) = (-3(-5(2|((1(-4x^2 + 4) - 3)/(-2(-4x^2 + 4) - 3)) - 1| - 2)^2 + 4(2|((1(-4x^2 + 4) - 3)/(-2(-4x^2 + 4) - 3)) - 1| - 2) + 2) - 2)/(-2(-5(2|((1(-4x^2 + 4) - 3)/(-2(-4x^2 + 4) - 3)) - 1| - 2)^2 + 4(2|((1(-4x^2 + 4) - 3)/(-2(-4x^2 + 4) - 3)) - 1| - 2) + 2) - 3), find the number... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(-3sin(1π(1(-1|(-1x - 3) - 3|) - 3) + 1)))/(3(-3sin(1π(1(-1|(-1x - 3) - 3|) - 3) + 1)) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -1|(-1|(-2(-4(2sin(1πx + 2) + 1)^2 + 1(2sin(1πx + 2) + 1) - 5) - 3) - 2| + 1) - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-5(-2cos(2π(-1((-3x - 2)/(-1x)) - 2) - 2) + 2)^2 - 2(-2cos(2π(-1((-3x - 2)/(-1x)) - 2) - 2) + 2) + 4) + 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 9 | MATH |
[
{
"content": "For the function f(x) = 1|(-4((-3(-3(3x^2 + 4x + 1) + 2) + 3)/(2(-3(3x^2 + 4x + 1) + 2) + 3))^2 - 1((-3(-3(3x^2 + 4x + 1) + 2) + 3)/(2(-3(3x^2 + 4x + 1) + 2) + 3)) + 4) + 2| + 1, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\b... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-2sin(2π(2(-2|(5x + 5) + 1| + 1) + 5) - 2) + 1)^2 + 5(-2sin(2π(2(-2|(5x + 5) + 1| + 1) + 5) - 2) + 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|((-3(3(-2cos(3π(1x^2 - 1x + 4) - 2) + 3)^2 + 4(-2cos(3π(1x^2 - 1x + 4) - 2) + 3) + 3) - 2)/(-2(3(-2cos(3π(1x^2 - 1x + 4) - 2) + 3)^2 + 4(-2cos(3π(1x^2 - 1x + 4) - 2) + 3) + 3) - 1)) - 2| + 1 negative?\n\nPresent... | 38 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1((-1(-4(1|(-1x^2 + 5x + 5) + 3| - 3)^2 - 4(1|(-1x^2 + 5x + 5) + 3| - 3) - 4) - 3)/(1(-4(1|(-1x^2 + 5x + 5) + 3| - 3)^2 - 4(1|(-1x^2 + 5x + 5) + 3| - 3) - 4) + 1)))/(-1((-1(-4(1|(-1x^2 + 5x + 5) + 3| - 3)^2... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(3(-5(3cos(3π((-3x - 1)/(-3x + 2)) - 2)) + 1)^2 + 3(-5(3cos(3π((-3x - 1)/(-3x + 2)) - 2)) + 1) + 3) - 1| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(-1(2sin(1π(-2|((3x - 1)/(-3x - 3)) - 1| - 2) - 3) + 1) - 5) - 3)/(1(-1(2sin(1π(-2|((3x - 1)/(-3x - 3)) - 1| - 2) - 3) + 1) - 5)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{You... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3π(1|(2((2(1|x - 3| - 1) + 1)/(-1(1|x - 3| - 1) + 2))) + 1| + 1) - 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3π(4((1(1|(-2x - 3) - 1| - 2) - 1)/(-1(1|(-2x - 3) - 1| - 2) - 2)) + 1) - 1) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "For the function f(x) = -1(2cos(3π(2|(-2(-3x + 3) - 4) - 2| - 1) + 2) - 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"
}
] | 38 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(3(2(1sin(3π(3|x + 3| - 2) - 1) + 1) - 4)) + 1)/(3(3(2(1sin(3π(3|x + 3| - 2) - 1) + 1) - 4)) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 91 | MATH |
[
{
"content": "For the function f(x) = -1(2(-2sin(2π(4((2x + 3)/(-2x + 3))^2 + 1((2x + 3)/(-2x + 3)) + 2) - 1) + 3)^2 + 3(-2sin(2π(4((2x + 3)/(-2x + 3))^2 + 1((2x + 3)/(-2x + 3)) + 2) - 1) + 3) - 1)^2 + 1(2(-2sin(2π(4((2x + 3)/(-2x + 3))^2 + 1((2x + 3)/(-2x + 3)) + 2) - 1) + 3)^2 + 3(-2sin(2π(4((2x + 3)/(-2x + 3... | 6 | MATH |
[
{
"content": "For the function f(x) = -1|(2(-3cos(3π((-3(-3|x - 2| - 2) - 2)/(1(-3|x - 2| - 2) - 3)) - 1) + 1) - 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"
}
] | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1((3(3cos(3π(-1|((2x - 2)/(-1x - 3)) - 2| - 2) + 3) + 3) - 3)/(-1(3cos(3π(-1|((2x - 2)/(-1x - 3)) - 2| - 2) + 3) + 3) - 1)) - 1)/(-2((3(3cos(3π(-1|((2x - 2)/(-1x - 3)) - 2| - 2) + 3) + 3) - 3)/(-1(3cos(3π(-1|((2x... | 19 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-2cos(3π(1(-3|(-4x^2 - 2x - 1) + 2|)^2 - 5) - 2)) - 1)/(-3(-2cos(3π(1(-3|(-4x^2 - 2x - 1) + 2|)^2 - 5) - 2))) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-4(-2(-3(-3cos(2πx - 2) + 1)^2 + 3(-3cos(2πx - 2) + 1) - 3) - 1) - 1) + 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -3sin(1π(1|(1(-1(-2x^2 - 1x + 2)^2 + 4(-2x^2 - 1x + 2) - 4)^2 - 3(-1(-2x^2 - 1x + 2)^2 + 4(-2x^2 - 1x + 2) - 4) - 1) + 1| - 1) + 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}"... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(2cos(3π((2((-3(-3x - 1) + 1)/(-3(-3x - 1) + 4)) - 2)/(2((-3(-3x - 1) + 1)/(-3(-3x - 1) + 4)) + 3)) + 1) + 3)^2 - 4(2cos(3π((2((-3(-3x - 1) + 1)/(-3(-3x - 1) + 4)) - 2)/(2((-3(-3x - 1) + 1)/(-3(-3x - 1) + 4... | 5 | MATH |
[
{
"content": "For the function f(x) = 2|(-2(-1sin(2π(-3|(-3x^2 - 5x - 2) + 2|) + 2) - 1)^2 + 5(-1sin(2π(-3|(-3x^2 - 5x - 2) + 2|) + 2) - 1) + 1) - 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"... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1π(-3(-3((-3(-4x^2 + 5x + 4))/(3(-4x^2 + 5x + 4) + 1)) - 5)^2 - 3) - 3) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1((-2(-1(2|(2cos(2πx) + 3) - 2| - 3) - 4) - 3)/(2(-1(2|(2cos(2πx) + 3) - 2| - 3) - 4) - 3)) + 3)/(-2((-2(-1(2|(2cos(2πx) + 3) - 2| - 3) - 4) - 3)/(2(-1(2|(2cos(2πx) + 3) - 2| - 3) - 4) - 3)) + 1) positive?\n\nPre... | 40 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(1(-1|(2sin(3π((3x + 2)/(2x - 1))) + 3) + 3| + 3) - 2) - 3)/(-3(1(-1|(2sin(3π((3x + 2)/(2x - 1))) + 3) + 3| + 3) - 2) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2π(2(-3|((1((-1x - 2)/(-3x - 1)) - 1)/(-1((-1x - 2)/(-3x - 1)) - 3)) + 2| + 2) - 1) + 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((2(-1|(4(-3|x - 2| + 2)^2 + 4(-3|x - 2| + 2) - 3) - 2| + 3) + 2)/(3(-1|(4(-3|x - 2| + 2)^2 + 4(-3|x - 2| + 2) - 3) - 2| + 3) + 2)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "For the function f(x) = 1|(-1|(2sin(3π(1(3|x| - 3)^2 + 1(3|x| - 3) + 5)) - 3) - 3| - 2) + 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"
}
] | 33 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2(-3sin(2π(-5(-4x + 2) + 1) + 2) - 3) + 5) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-2cos(2π(1((3((1x)/(-1x + 2)) + 2)/(1((1x)/(-1x + 2))))^2 - 5((3((1x)/(-1x + 2)) + 2)/(1((1x)/(-1x + 2)))) - 4)) + 3) + 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 15 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((3((-1(2sin(2π(1|x - 2|) - 1) + 3) - 2)/(-2(2sin(2π(1|x - 2|) - 1) + 3) - 3)))/(-2((-1(2sin(2π(1|x - 2|) - 1) + 3) - 2)/(-2(2sin(2π(1|x - 2|) - 1) + 3) - 3)) - 1)) + 3| is strictly increasing.\n\nPresent... | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(1sin(3π((-1(-2|((-1x - 1)/(3x - 1)) - 1| + 3))/(1(-2|((-1x - 1)/(3x - 1)) - 1| + 3) - 3)))) - 2| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(3sin(3π(-2(-2(-4x^2 + 1)^2 - 3(-4x^2 + 1) - 4)) - 3) + 2) + 3| - 1 positive?\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) = 3cos(3π(-1|(-4(-2|(-1x - 3) - 1|) - 3) - 3| - 2) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(3sin(3π(-2|(1(2|x + 2|) - 2) + 1| - 2) + 3) + 1) + 3)/(-1(3sin(3π(-2|(1(2|x + 2|) - 2) + 1| - 2) + 3) + 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 44 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-3|(-1(1sin(3π(3x - 1) + 2) - 2)^2 + 2(1sin(3π(3x - 1) + 2) - 2) - 1) - 3| - 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-3(-1cos(3π(3(1|x - 3|)) + 1) + 3)^2 - 4(-1cos(3π(3(1|x - 3|)) + 1) + 3) + 2) - 3| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(1π(1(2|(1|(5x^2 + 5x - 5)| - 3) - 2| - 2) + 1) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2π(-1((-3(1(-1x^2 - 3x + 4)^2 + 2(-1x^2 - 3x + 4) + 5) + 2)/(-3(1(-1x^2 - 3x + 4)^2 + 2(-1x^2 - 3x + 4) + 5) - 3)) + 3) + 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-1(-3sin(2π(3|(2|x| + 2)| - 3) + 3) - 3) - 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(-3(-2cos(1π(3|x|) - 2)) + 5)^2 + 3(-3(-2cos(1π(3|x|) - 2)) + 5) - 5)^2 - 1(1(-3(-2cos(1π(3|x|) - 2)) + 5)^2 + 3(-3(-2cos(1π(3|x|) - 2)) + 5) - 5) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5((2(-3(-2|(-1x^2 - 1x + 2) + 2| + 2)^2 - 3(-2|(-1x^2 - 1x + 2) + 2| + 2) + 4) - 3)/(-2(-3(-2|(-1x^2 - 1x + 2) + 2| + 2)^2 - 3(-2|(-1x^2 - 1x + 2) + 2| + 2) + 4) - 3)) - 5 is strictly increasing.\n\nPresent ... | 4 | MATH |
[
{
"content": "For the function f(x) = 5((3((-2(5(3sin(3πx) - 3)^2 + 1(3sin(3πx) - 3) - 1) - 3)/(-1(5(3sin(3πx) - 3)^2 + 1(3sin(3πx) - 3) - 1) + 2)) - 1)/(-1((-2(5(3sin(3πx) - 3)^2 + 1(3sin(3πx) - 3) - 1) - 3)/(-1(5(3sin(3πx) - 3)^2 + 1(3sin(3πx) - 3) - 1) + 2)))), find the number of maximal connected intervals ... | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(1sin(3π((1(4(1|x + 3| - 3)^2 - 2(1|x + 3| - 3)) + 2)/(3(4(1|x + 3| - 3)^2 - 2(1|x + 3| - 3)) + 2)) + 2) - 1)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2π(4((2(3(1|x - 2| - 3)^2 - 5(1|x - 2| - 3) + 4) - 1)/(-1(3(1|x - 2| - 3)^2 - 5(1|x - 2| - 3) + 4) - 2)) - 1) - 2) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(1π(3(-2|((3(-4x^2 - 3x - 5))/(-3(-4x^2 - 3x - 5) + 2)) + 3| - 2)^2 + 3(-2|((3(-4x^2 - 3x - 5))/(-3(-4x^2 - 3x - 5) + 2)) + 3| - 2) + 1)) + 2 is strictly increasing.\n\nPresent the answer in LaTex format... | 10 | MATH |
[
{
"content": "For the function f(x) = 3(1|(3(-3sin(1π((-2x + 2)/(-1x - 2)) + 1) + 3)^2 - 3(-3sin(1π((-2x + 2)/(-1x - 2)) + 1) + 3) - 5) + 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"
}... | 11 | MATH |
[
{
"content": "For the function f(x) = 2(-1(-1sin(2π(2|(1x + 3) + 3| - 1) - 2) - 3)^2 + 5(-1sin(2π(2|(1x + 3) + 3| - 1) - 2) - 3) + 4) + 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"
}
] | 40 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-5(4(-3|(2sin(2πx + 3) - 2) + 1| + 2)^2 + 4(-3|(2sin(2πx + 3) - 2) + 1| + 2) + 1) - 1) - 2)/(3(-5(4(-3|(2sin(2πx + 3) - 2) + 1| + 2)^2 + 4(-3|(2sin(2πx + 3) - 2) + 1| + 2) + 1) - 1) + 1) is strictly decre... | 55 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2((1((-1(1cos(3π(-2|x - 3| + 1) + 3)) + 1)/(3(1cos(3π(-2|x - 3| + 1) + 3)) + 3)) + 1)/(-1((-1(1cos(3π(-2|x - 3| + 1) + 3)) + 1)/(3(1cos(3π(-2|x - 3| + 1) + 3)) + 3)) + 3)) + 3)/(1((1((-1(1cos(3π(-2|x - 3| + 1) + ... | 61 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(1(-2sin(3π(-4((-2x - 2)/(-2x - 1)) - 5) + 3)) + 5) - 1)/(1(1(-2sin(3π(-4((-2x - 2)/(-2x - 1)) - 5) + 3)) + 5) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(5(5(-3sin(3π(-1|x + 1| + 1) - 1) - 3) - 1) + 2) + 2)/(2(5(5(-3sin(3π(-1|x + 1| + 1) - 1) - 3) - 1) + 2) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 31 | MATH |
[
{
"content": "For the function f(x) = -5(5(-2|((-3((-2x + 1)/(-2x - 3)) - 2)/(2((-2x + 1)/(-2x - 3)) - 3)) + 1| + 1)^2 - 5(-2|((-3((-2x + 1)/(-2x - 3)) - 2)/(2((-2x + 1)/(-2x - 3)) - 3)) + 1| + 1) - 3)^2 - 2(5(-2|((-3((-2x + 1)/(-2x - 3)) - 2)/(2((-2x + 1)/(-2x - 3)) - 3)) + 1| + 1)^2 - 5(-2|((-3((-2x + 1)/(-2x... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2((3(3(-1cos(2π(2x^2 + 4) + 3) + 2) + 2) + 2)/(2(3(-1cos(2π(2x^2 + 4) + 3) + 2) + 2) + 3)) - 3)/(3((3(3(-1cos(2π(2x^2 + 4) + 3) + 2) + 2) + 2)/(2(3(-1cos(2π(2x^2 + 4) + 3) + 2) + 2) + 3)) + 1) negative?\n\nPrese... | 51 | MATH |
[
{
"content": "For the function f(x) = -5(2|((-1(-2|(2sin(2πx) + 3) + 1| + 2) + 1)/(-1(-2|(2sin(2πx) + 3) + 1| + 2)))| - 1)^2 - 2(2|((-1(-2|(2sin(2πx) + 3) + 1| + 2) + 1)/(-1(-2|(2sin(2πx) + 3) + 1| + 2)))| - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answ... | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2(-1cos(2π(-3(-5x + 5)^2 - 2(-5x + 5) - 3) - 3) - 2) - 2)^2 - 4(-2(-1cos(2π(-3(-5x + 5)^2 - 2(-5x + 5) - 3) - 3) - 2) - 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 7 | MATH |
[
{
"content": "For the function f(x) = (-1(-2(2|(5(-4x^2 + 2x + 3)^2 + 1(-4x^2 + 2x + 3) - 2) - 3|) - 4) - 1)/(3(-2(2|(5(-4x^2 + 2x + 3)^2 + 1(-4x^2 + 2x + 3) - 2) - 3|) - 4) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 4 | MATH |
[
{
"content": "For the function f(x) = 2(-1(-2sin(2π(-1|(-1x^2 + 1x - 4)| + 2) - 2) - 1)^2 + 1(-2sin(2π(-1|(-1x^2 + 1x - 4)| + 2) - 2) - 1) + 5) - 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"
}
] | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-3(3(-1cos(2π(4x^2 + 2x + 3) + 2) - 1)^2 - 4) + 1)^2 + 1(-3(3(-1cos(2π(4x^2 + 2x + 3) + 2) - 1)^2 - 4) + 1) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = 5(-5(-3cos(2π(2(1x - 5)^2 - 1(1x - 5) - 5)) - 1) + 4), 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) = 4((-2((3(2cos(3π(-3x - 1) + 2) - 3))/(-3(2cos(3π(-3x - 1) + 2) - 3) + 4)) + 1)/(-1((3(2cos(3π(-3x - 1) + 2) - 3))/(-3(2cos(3π(-3x - 1) + 2) - 3) + 4)) - 1))^2 + 3((-2((3(2cos(3π(-3x - 1) + 2) - 3))/(-3(2cos(... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1(1cos(3π(3|(-1|x - 2| + 2) + 2| - 1) - 3) + 1)^2 - 3(1cos(3π(3|(-1|x - 2| + 2) + 2| - 1) - 3) + 1) + 5)^2 - 1(1(1cos(3π(3|(-1|x - 2| + 2) + 2| - 1) - 3) + 1)^2 - 3(1cos(3π(3|(-1|x - 2| + 2) + 2| - 1) - 3) + 1)... | 46 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(4((2(-2(1cos(3πx - 1) - 3)^2 - 2(1cos(3πx - 1) - 3) - 5) - 1)/(-2(-2(1cos(3πx - 1) - 3)^2 - 2(1cos(3πx - 1) - 3) - 5) - 3))^2 - 1((2(-2(1cos(3πx - 1) - 3)^2 - 2(1cos(3πx - 1) - 3) - 5) - 1)/(-2(-2(1cos(3πx... | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(4(2e^(-2(-1(2|x - 1| - 1)^2 - 1(2|x - 1| - 1))) - 2)^2 + 4(2e^(-2(-1(2|x - 1| - 1)^2 - 1(2|x - 1| - 1))) - 2) - 3) - 1)/(1(4(2e^(-2(-1(2|x - 1| - 1)^2 - 1(2|x - 1| - 1))) - 2)^2 + 4(2e^(-2(-1(2|x - 1| - 1)^2 - ... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(4(1|(-2(-2cos(3πx - 3) + 1)^2 - 3(-2cos(3πx - 3) + 1) + 4) - 1| - 3) + 4) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(3|(3(3sin(3π(-1x + 4)) + 1) - 2) + 2|) - 2| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((3(-3cos(2π((-3((3x)/(-2x - 1)) - 1)/(2((3x)/(-2x - 1)) + 2)) + 2)) - 2)/(3(-3cos(2π((-3((3x)/(-2x - 1)) - 1)/(2((3x)/(-2x - 1)) + 2)) + 2)) - 1))^2 + 2((3(-3cos(2π((-3((3x)/(-2x - 1)) - 1)/(2((3x)/(-2x - ... | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-2(2cos(3π((-1(1|(-5x^2 + 2x + 4) + 2| - 2) + 3)/(2(1|(-5x^2 + 2x + 4) + 2| - 2) - 1)) + 3) - 2) + 2)/(2(2cos(3π((-1(1|(-5x^2 + 2x + 4) + 2| - 2) + 3)/(2(1|(-5x^2 + 2x + 4) + 2| - 2) - 1)) + 3) - 2)) positive?\n\... | 9 | MATH |
[
{
"content": "For the function f(x) = 4(-1sin(3π((-3((-2((-1x - 3)/(1x + 1)) + 1)/(1((-1x - 3)/(1x + 1)) + 1)) + 1)/(-2((-2((-1x - 3)/(1x + 1)) + 1)/(1((-1x - 3)/(1x + 1)) + 1)) + 3)) + 3) + 2) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex forma... | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3π(2(-1(3|(-1|x + 2| - 3)| + 2) - 4)) + 1) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -1|(-2|((1((2(-1sin(3πx) - 3) - 1)/(1(-1sin(3πx) - 3) - 2)) + 2)/(-2((2(-1sin(3πx) - 3) - 1)/(1(-1sin(3πx) - 3) - 2)) - 1)) + 2| + 3) - 3| - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 30 | MATH |
[
{
"content": "For the function f(x) = 1(-3(-1sin(3π(2|(1x - 2) - 2| - 1) + 2) - 3)^2 - 4(-1sin(3π(2|(1x - 2) - 2| - 1) + 2) - 3) + 5) + 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"
}
] | 61 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3((3(-1sin(3π(3|(-2x^2 + 4x + 5) + 1| + 1) - 1) + 2) + 1)/(-2(-1sin(3π(3|(-2x^2 + 4x + 5) + 1| + 1) - 1) + 2) - 2)) - 2)/(-2((3(-1sin(3π(3|(-2x^2 + 4x + 5) + 1| + 1) - 1) + 2) + 1)/(-2(-1sin(3π(3|(-2x^2 + 4x + 5)... | 47 | MATH |
[
{
"content": "For the function f(x) = 4(2sin(3π((1(3|(-2|x - 2|) + 3| - 3) + 3)/(-2(3|(-2|x - 2|) + 3| - 3) + 3)) - 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"
}
] | 15 | MATH |
[
{
"content": "For the function f(x) = 3|(1|(1|(-3sin(2π(-1x - 3) - 2) - 1) - 3| - 1) + 1| - 3) + 3| - 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"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-3|(1(1(2cos(1πx + 2) + 3)^2 - 4(2cos(1πx + 2) + 3)) + 5) - 1| - 2) - 2)/(-2(-3|(1(1(2cos(1πx + 2) + 3)^2 - 4(2cos(1πx + 2) + 3)) + 5) - 1| - 2) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(3|(-1(-4(3sin(2πx - 3) + 2) + 3)^2 + 3(-4(3sin(2πx - 3) + 2) + 3) - 4) - 2| + 1) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "For the function f(x) = (2((1(-3|(-1(-1cos(2πx + 3))^2 - 3(-1cos(2πx + 3))) + 3| - 2) - 1)/(1(-3|(-1(-1cos(2πx + 3))^2 - 3(-1cos(2πx + 3))) + 3| - 2) + 2)) + 1)/(-1((1(-3|(-1(-1cos(2πx + 3))^2 - 3(-1cos(2πx + 3))) + 3| - 2) - 1)/(1(-3|(-1(-1cos(2πx + 3))^2 - 3(-1cos(2πx + 3))) + 3| - 2) + 2)) + 3)... | 30 | MATH |
[
{
"content": "For the function f(x) = 3sin(2π(-2|(1((2(-1x - 3) - 1)/(1(-1x - 3))) - 3) - 3| + 2) + 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"
}
] | 10 | MATH |
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