messages listlengths 1 1 | ground_truth stringclasses 81 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "For the function f(x) = -2|(3(-3sin(2πx + 3) - 1) - 5) + 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"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = (-1(1cos(2π(1|(-5x^2 - 1x) + 1| - 2)) + 1) - 2)/(1(1cos(2π(1|(-5x^2 - 1x) + 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"
}
] | 52 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(2sin(3π(-1x - 4) - 3) + 3) - 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "For the function f(x) = -2sin(1π(4((-2x)/(-1x - 3))^2 + 3((-2x)/(-1x - 3)) - 3) + 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}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "For the function f(x) = 1cos(2π(-2|(1x + 1) + 1| + 3) - 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 answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2cos(1π((2x)/(-1x - 2)) - 1)) + 2| - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(1sin(3π(-1|x + 3| - 2)) - 3)^2 + 4(1sin(3π(-1|x + 3| - 2)) - 3) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = -2|((-3(1sin(3π(2x - 2) - 2) + 1) + 3)/(-2(1sin(3π(2x - 2) - 2) + 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"
}
] | 62 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((2(-1sin(1πx + 3) + 2))/(-2(-1sin(1πx + 3) + 2) + 3))^2 + 4((2(-1sin(1πx + 3) + 2))/(-2(-1sin(1πx + 3) + 2) + 3)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 21 | MATH |
[
{
"content": "For the function f(x) = 3cos(3π(-3((-3(2|x + 2| + 2) - 1)/(-1(2|x + 2| + 2) - 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: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-3cos(3π(-5(-4x^2 + 3x - 3) - 2) - 3) - 2) - 1| + 2 is strictly decreasing.\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) = (-3(-2sin(1π(3x - 5) - 1) + 1) + 3)/(-2(-2sin(1π(3x - 5) - 1) + 1) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = 5(1sin(3π(-3|((-3x + 2)/(2x + 3)) + 2| + 3) + 3))^2 - 3(1sin(3π(-3|((-3x + 2)/(2x + 3)) + 2| + 3) + 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"
}
] | 27 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((-2(-1cos(1π(2|x - 2| - 1) - 3) - 1) - 3)/(1(-1cos(1π(2|x - 2| - 1) - 3) - 1) + 2)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-1(-2sin(1π(2x - 5) + 1) - 1) - 2)/(3(-2sin(1π(2x - 5) + 1) - 1) + 1))^2 + 2((-1(-2sin(1π(2x - 5) + 1) - 1) - 2)/(3(-2sin(1π(2x - 5) + 1) - 1) + 1)) + 3 is strictly decreasing.\n\nPresent the answer in La... | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(3π(4(2|x| - 2) - 1) + 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-4(1sin(2πx - 3) + 2) + 2)^2 - 4(-4(1sin(2πx - 3) + 2) + 2) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -1(1cos(2π(1((-2x + 1)/(2x + 1)) + 5) - 1) - 1)^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"
}
] | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2π(5(1|x - 1| + 1)^2 - 3(1|x - 1| + 1) - 1) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "For the function f(x) = (1(-2cos(1π(5x + 2) - 2) + 3) - 2)/(3(-2cos(1π(5x + 2) - 2) + 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"
}
] | 50 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2sin(3π(-4((-1x + 1)/(3x - 2)) + 2) - 2) - 2 is strictly increasing.\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) = 1sin(1π(-2|(-1x + 5) - 1| - 2)) + 1 negative?\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) = 1cos(3π(1|(5(4x^2 - 2x + 5) + 5) + 1| - 2) + 2) + 2 positive?\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|(-2cos(1π(-4x^2 - 5x) - 1) + 3) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "For the function f(x) = -5(-3cos(2π(1|x - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2(-2cos(1πx - 3) + 3) + 2) - 2)/(-3(-2(-2cos(1πx - 3) + 3) + 2)) is strictly decreasing.\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(3|(-1sin(2πx + 3) - 1) - 2|) - 5 negative?\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) = -3sin(1π(1(1((3x - 3)/(-1x + 3)) + 1)^2 - 5(1((3x - 3)/(-1x + 3)) + 1) + 5) - 1) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(1|(-5(-2cos(1πx + 3))^2 + 4(-2cos(1πx + 3)) + 3) + 3|))/(3(1|(-5(-2cos(1πx + 3))^2 + 4(-2cos(1πx + 3)) + 3) + 3|) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 40 | MATH |
[
{
"content": "For the function f(x) = 2|((-2(3cos(1πx + 3) + 3) - 1)/(-1(3cos(1πx + 3) + 3) - 2)) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-4(1cos(3π(-3x - 3) - 2))^2 - 5(1cos(3π(-3x - 3) - 2)) - 1) - 2| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "For the function f(x) = 1|(-3sin(1π(2x - 5) + 3) + 2) - 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"
}
] | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-2cos(3π((1(2|x + 2|) - 3)/(2(2|x + 2|) - 1)) + 3) + 1)^2 + 5(-2cos(3π((1(2|x + 2|) - 3)/(2(2|x + 2|) - 1)) + 3) + 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 8 | MATH |
[
{
"content": "For the function f(x) = 1sin(3π((-3(-4x^2 - 4x - 3) - 1)/(1(-4x^2 - 4x - 3) - 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"
}
] | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-3(-2sin(2π(4x)) + 3)^2 + 5(-2sin(2π(4x)) + 3) - 1) + 2)/(-3(-3(-2sin(2π(4x)) + 3)^2 + 5(-2sin(2π(4x)) + 3) - 1) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 80 | MATH |
[
{
"content": "For the function f(x) = (2(-3cos(3π(1|x + 1| + 2) + 2) + 3))/(-2(-3cos(3π(1|x + 1| + 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"
}
] | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((2(4(-2cos(3πx + 3) + 3) + 4) + 3)/(1(4(-2cos(3πx + 3) + 3) + 4))) + 2| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = 2sin(3π(1((3x + 3)/(-3x + 1))^2 + 5((3x + 3)/(-3x + 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}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-1|(2sin(3πx + 1) - 3) + 3| - 2) + 2)/(-3(-1|(2sin(3πx + 1) - 3) + 3| - 2) + 2) 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) = 5(-1sin(2π(-1|x + 3| + 1) + 2) + 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = 3(-5(-1cos(2πx - 2) - 3) + 1)^2 - 3(-5(-1cos(2πx - 2) - 3) + 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}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(3((3(3sin(1πx - 2) + 3) - 3)/(-3(3sin(1πx - 2) + 3))) - 4)^2 + 5(3((3(3sin(1πx - 2) + 3) - 3)/(-3(3sin(1πx - 2) + 3))) - 4) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(2π(3(-2|x + 1|)^2 - 1(-2|x + 1|) - 3) - 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = -4((-1(-3sin(3π(-3|x + 2| + 2))) + 2)/(3(-3sin(3π(-3|x + 2| + 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}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((1(-2cos(1πx + 3) + 1) + 1)/(-2(-2cos(1πx + 3) + 1) + 3)) + 3 is strictly decreasing.\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|(5(3sin(3πx - 2) - 2)^2 - 5(3sin(3πx - 2) - 2) - 3) + 1| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2π(-4((3(-2|x| + 3) - 2)/(1(-2|x| + 3) + 2)) - 5) - 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((3(3cos(3π(4x^2 + 5x - 3) + 2) - 3) - 1)/(2(3cos(3π(4x^2 + 5x - 3) + 2) - 3))) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2π(-5(1(2|x| - 3)^2 + 1(2|x| - 3) + 3) + 1) - 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2|((-2(-2cos(3πx - 2) + 2) - 2)/(-3(-2cos(3πx - 2) + 2) - 2)) - 1| - 3) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1(1sin(1π(1|x - 3|) - 3) + 2) - 3)^2 + 4(-1(1sin(1π(1|x - 3|) - 3) + 2) - 3) + 2 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) = -5(-3cos(3π(1((3x - 3)/(-1x + 2))^2 - 3((3x - 3)/(-1x + 2)) - 2) + 3) + 1) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-3|(-3(3cos(1πx - 1) + 2)^2 - 5)| - 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -2cos(3π((-2(2|(2x^2 - 5x) - 1|) + 1)/(1(2|(2x^2 - 5x) - 1|) - 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"
}
] | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2|((-3(2cos(3πx - 3) - 3) + 2)/(-2(2cos(3πx - 3) - 3) - 2)) - 3| + 3)^2 + 3(2|((-3(2cos(3πx - 3) - 3) + 2)/(-2(2cos(3πx - 3) - 3) - 2)) - 3| + 3) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your... | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-2|(2cos(1πx + 1) - 3)| + 3) + 2)/(-1(-2|(2cos(1πx + 1) - 3)| + 3) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|((-1(1(2cos(1πx - 1) + 1)^2 + 3) + 1)/(-3(1(2cos(1πx - 1) + 1)^2 + 3) + 2)) + 1| + 2 is strictly decreasing.\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((-2(-2sin(1πx) + 3) + 2)/(-2(-2sin(1πx) + 3) - 1)) - 1 negative?\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) = -1sin(3π(-2(-2|x - 2| - 2) + 3)) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 85 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(5(-1sin(2π(-3|x + 1| + 1) - 2) - 3)^2 - 3(-1sin(2π(-3|x + 1| + 1) - 2) - 3) - 2))/(2(5(-1sin(2π(-3|x + 1| + 1) - 2) - 3)^2 - 3(-1sin(2π(-3|x + 1| + 1) - 2) - 3) - 2) + 2) positive?\n\nPresent the answer in LaT... | 61 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(4((2(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2) - 3))^2 + 1((2(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2) - 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 answer}",
"role": "user"
}... | 27 | MATH |
[
{
"content": "For the function f(x) = 1cos(1π(2(-1(-1|x + 3| + 3))^2 - 4(-1(-1|x + 3| + 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"
}
] | 67 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(3π(-2(2|x + 1| + 1) + 5) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-2(-1cos(1π(-3x^2 + 4x + 2) + 1)) + 1)/(-1(-1cos(1π(-3x^2 + 4x + 2) + 1)) - 3)) - 1| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1((-1(-1sin(3π(4x + 1) - 3)))/(1(-1sin(3π(4x + 1) - 3)) + 3))^2 - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1(2cos(3π(-2x^2 - 4x - 1) + 1) - 1) + 4)| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(2cos(2π(4x^2 + 3) + 3) + 2) + 3| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(4(-2|(-2cos(3πx - 2) + 3) + 3| + 3)^2 + 5(-2|(-2cos(3πx - 2) + 3) + 3| + 3) + 2) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3((-3(-1cos(3π(2|x - 1| - 3) - 3) - 2) + 2)/(3(-1cos(3π(2|x - 1| - 3) - 3) - 2) + 3))^2 - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((-1(3|(-2sin(2πx + 1) + 2)| + 3))/(2(3|(-2sin(2πx + 1) + 2)| + 3) + 1)) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = 2((1(-3sin(1πx + 1) - 3) - 1)/(-1(-3sin(1πx + 1) - 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 answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1π(1|((-3(-1x^2 + 3) + 2)/(-1(-1x^2 + 3) + 2)) + 2| + 1) + 3) - 3 is strictly decreasing.\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|(1cos(2π(2x - 2) + 1) - 3) + 2| - 3 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) = 3cos(3π((-1(-1(-2|x - 1| + 1)) + 3)/(3(-1(-2|x - 1| + 1)) + 2))) + 3 positive?\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) = 2|(3sin(2π(2x + 4) - 2) - 1) - 2| + 3 positive?\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(-1cos(3π(2|(-4x^2 + 1x - 4) - 2| - 1) - 3) - 2) - 1)/(-2(-1cos(3π(2|(-4x^2 + 1x - 4) - 2| - 1) - 3) - 2) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(1sin(2π(1(1x^2 + 3) - 5) - 2) + 3) - 3| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2sin(2π((1x + 2)/(-2x - 1)) + 3) + 3) + 2| + 1 is strictly increasing.\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) = 3cos(3π((-3(1(-2x + 2)^2 - 5(-2x + 2) + 2) - 2)/(1(1(-2x + 2)^2 - 5(-2x + 2) + 2) + 3)) - 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-3sin(2π(-5x^2 + 1x)) - 3) - 2| - 1 is strictly increasing.\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) = -2|(-3(-2sin(3π((2x + 1)/(-2x + 3)) + 3) + 3)^2 + 4(-2sin(3π((2x + 1)/(-2x + 3)) + 3) + 3)) - 2| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = 2|(-5(5(1cos(1πx - 2) + 1) - 4)^2 + 2(5(1cos(1πx - 2) + 1) - 4) + 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"
}
] | 26 | MATH |
[
{
"content": "For the function f(x) = 5(2sin(2π(1((-1x + 1)/(-1x))^2 - 2((-1x + 1)/(-1x)) - 3) + 1) - 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}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1sin(3π(-4x^2 + 1) + 3) - 1) - 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-5(1|(1cos(2πx) + 2)| + 2) - 3) - 1)/(2(-5(1|(1cos(2πx) + 2)| + 2) - 3) - 3) negative?\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) = 1(1(-2sin(2πx - 3) + 2)^2 + 5(-2sin(2πx - 3) + 2) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -2|(2cos(2π(5((3x - 1)/(-2x + 1))^2 - 4((3x - 1)/(-2x + 1))) - 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"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2sin(1π(-2((-3x - 3)/(3x + 2))^2 + 3((-3x - 3)/(3x + 2)) + 4) - 1) - 1) + 2 is strictly decreasing.\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) = 2sin(2π((-2(5(-1|x - 1| + 2)^2 - 4(-1|x - 1| + 2) + 2) - 1)/(1(5(-1|x - 1| + 2)^2 - 4(-1|x - 1| + 2) + 2) + 2)) - 3) - 1 positive?\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) = -2sin(2π(-3(-1((-3x + 1)/(-1x))^2 + 5((-3x + 1)/(-1x)) + 4) + 1) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = -2|(-4((-2(2cos(1πx - 1) - 1) - 2)/(-1(2cos(1πx - 1) - 1) + 2))^2 + 5((-2(2cos(1πx - 1) - 1) - 2)/(-1(2cos(1πx - 1) - 1) + 2)) - 4) + 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 ans... | 21 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π((2(-3(2|x + 2| - 3)^2 - 3(2|x + 2| - 3) - 5) + 2)/(-1(-3(2|x + 2| - 3)^2 - 3(2|x + 2| - 3) - 5) - 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": "us... | 10 | MATH |
[
{
"content": "For the function f(x) = 3|(4(-2sin(3π(-3x + 3) + 3) - 1)^2 + 4(-2sin(3π(-3x + 3) + 3) - 1) - 2) - 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"
}
] | 47 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(2(1sin(1πx + 1) + 3) - 4) - 2| - 2 is strictly increasing.\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) = 4(-1cos(3π(-5x^2 - 1) - 1) - 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "For the function f(x) = 5(-2cos(3π((1x + 3)/(3x - 2)) - 1) - 1)^2 + 5(-2cos(3π((1x + 3)/(3x - 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"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -4(-5(2sin(1πx - 2) - 1) - 4)^2 - 1(-5(2sin(1πx - 2) - 1) - 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"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(3π(4(-4x - 5)^2 - 5(-4x - 5) - 5) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "For the function f(x) = (-1(3(3cos(3πx + 3) + 3) - 4) - 1)/(-3(3(3cos(3πx + 3) + 3) - 4) + 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"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = -2(-3|((3(-1sin(3πx + 1) - 2))/(-2(-1sin(3πx + 1) - 2) - 3)) - 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"
}
] | 43 | MATH |
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