messages listlengths 1 1 | ground_truth stringclasses 81
values | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1sin(3π(3x^2 - 3x - 3) + 1) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(3|(2cos(2πx - 2) - 1) + 3| + 3) + 1 is strictly decreasing.\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(-2cos(3π(-1((1x + 1)/(2x + 3))^2 - 1((1x + 1)/(2x + 3)) + 5) - 1)) + 5 positive?\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) = -3sin(1π(-3|((-1x + 3)/(2x - 1))| - 1) + 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(1(-3sin(2πx - 3) - 3) + 3) - 1)/(3(1(-3sin(2πx - 3) - 3) + 3) - 2) negative?\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) = -2(2((2(-3sin(2πx + 2) - 1) - 3)/(2(-3sin(2πx + 2) - 1) + 1)))^2 + 5(2((2(-3sin(2πx + 2) - 1) - 3)/(2(-3sin(2πx + 2) - 1) + 1))) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{You... | 40 | MATH |
[
{
"content": "For the function f(x) = 3(-3(-1cos(3πx + 1) + 2) + 1)^2 + 2(-3(-1cos(3πx + 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"
}
] | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3sin(1π(-3(3|x + 1| - 2) - 5) - 3) + 1) - 1)/(-2(3sin(1π(-3(3|x + 1| - 2) - 5) - 3) + 1)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 84 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(-3|(-1(-1cos(2πx - 3) + 1)^2 + 2(-1cos(2πx - 3) + 1) + 2) + 3| + 1) - 2 negative?\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) = -1sin(1π(4(-2|x| + 2)^2 + 4(-2|x| + 2) - 4) + 2) is strictly decreasing.\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) = 1|(-1(1cos(3πx + 2) - 1) - 1) - 1| + 3 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) = -5(1cos(3π(2|x - 1| + 1)) + 3) - 3 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) = (-3(-2cos(2π(-2(1|x - 1| + 2)^2 - 2(1|x - 1| + 2) + 2) + 2)) + 3)/(3(-2cos(2π(-2(1|x - 1| + 2)^2 - 2(1|x - 1| + 2) + 2) + 2)) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 49 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3π(1(1|x - 2| - 1) + 2) + 1) - 2 is strictly decreasing.\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) = 1(1sin(2π(-5(3|x - 2| + 2) - 5)) - 2)^2 - 1(1sin(2π(-5(3|x - 2| + 2) - 5)) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 105 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1π(-2(3|((-2x + 3)/(2x - 1)) + 2| + 1) - 1) - 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -3cos(1π(-1|((3(2x + 1) - 1)/(-1(2x + 1) + 2)) + 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"
}
] | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-2(2cos(3π(1|x| + 2) - 3) + 1) - 3)^2 + 5(-2(2cos(3π(1|x| + 2) - 3) + 1) - 3) - 5 is strictly decreasing.\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) = 3sin(1π(5(-1|x - 1| + 3)^2) - 3) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "For the function f(x) = -2(-2cos(1π(-3|x - 1| + 3) - 1) - 2)^2 - 5(-2cos(1π(-3|x - 1| + 3) - 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"
}
] | 60 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-2(-2sin(3πx - 1) + 2) - 2)| + 1 negative?\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) = (1(1|(5(2sin(3πx - 2)) + 2) - 3| + 3))/(-3(1|(5(2sin(3πx - 2)) + 2) - 3| + 3) + 1) is strictly increasing.\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(-2sin(2π(2(-3x - 1)^2 + 5(-3x - 1) - 2) + 2) + 3))/(-3(-2sin(2π(2(-3x - 1)^2 + 5(-3x - 1) - 2) + 2) + 3) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(1π(4(1|x + 1|)) + 2) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = (-2(-2cos(2π(-3x^2 - 3x - 5) + 1) - 3) - 2)/(1(-2cos(2π(-3x^2 - 3x - 5) + 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"
}
] | 49 | MATH |
[
{
"content": "For the function f(x) = 4(2sin(2π((-1x - 3)/(-3x + 1)) + 3) - 1)^2 - 2(2sin(2π((-1x - 3)/(-3x + 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"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -1((-1(1|(-1sin(2πx - 3) - 2) - 2| - 1) - 1)/(2(1|(-1sin(2πx - 3) - 2) - 2| - 1) - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(3π(-3(2|(-4x^2 - 3x + 1) + 3| - 3) - 5) - 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1(-1cos(1πx + 1) + 2)^2 + 1(-1cos(1πx + 1) + 2) - 2) + 3| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = 1(1sin(2π(1(-2|x + 3| + 3) - 3) + 2) - 1)^2 + 5(1sin(2π(1(-2|x + 3| + 3) - 3) + 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"
}
] | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(4(3sin(3πx + 3) - 3)^2 + 1(3sin(3πx + 3) - 3) - 5) - 2 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) = 1|(-1(-2sin(2πx - 3)) + 5) + 3| - 1 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) = -1|(2cos(2π(-1x + 3) - 1) - 1) - 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -4((3(-2|(-1cos(1πx - 2) + 1) + 2| + 1) - 3)/(2(-2|(-1cos(1πx - 2) + 1) + 2| + 1) - 1))^2 - 4((3(-2|(-1cos(1πx - 2) + 1) + 2| + 1) - 3)/(2(-2|(-1cos(1πx - 2) + 1) + 2| + 1) - 1)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent th... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3π(-3(3|(-4x^2 + 5x - 5) + 3| + 2) + 4)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(2(1sin(2πx + 3) + 2) + 1) + 1)/(-3(2(1sin(2πx + 3) + 2) + 1) + 4) 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) = (-1(-2(3(-2sin(1πx - 3) - 1)^2 + 3(-2sin(1πx - 3) - 1)) - 5) + 3)/(3(-2(3(-2sin(1πx - 3) - 1)^2 + 3(-2sin(1πx - 3) - 1)) - 5) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 20 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π(5(-1|(-3x^2 + 4x + 5)| - 3) - 1) + 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"
}
] | 44 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-1|(-1sin(2πx + 1) - 2) - 1| - 3)^2 - 5(-1|(-1sin(2πx + 1) - 2) - 1| - 3) + 2 negative?\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(3|(-2cos(3π(-4x - 1)) - 2) + 1| - 1) - 3)/(2(3|(-2cos(3π(-4x - 1)) - 2) + 1| - 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "For the function f(x) = -1|(-3sin(3π(-1(4x^2 - 1x + 4) + 5) + 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"
}
] | 32 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(4(2cos(2πx + 2)) + 3)^2 + 3(4(2cos(2πx + 2)) + 3) - 3 is strictly decreasing.\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) = (3(1(3cos(1π(4x + 2) - 1) + 1)^2 - 3(3cos(1π(4x + 2) - 1) + 1) - 5) - 3)/(1(1(3cos(1π(4x + 2) - 1) + 1)^2 - 3(3cos(1π(4x + 2) - 1) + 1) - 5) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format... | 52 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(-1(1cos(3πx + 2) - 2) - 1) + 2| + 3 is strictly increasing.\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) = -1sin(2π(4(5x + 2)^2 + 4(5x + 2) - 4) - 1) + 2 is strictly increasing.\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) = -1|(2(2(3sin(2πx - 1) - 1) - 4)^2 - 1(2(3sin(2πx - 1) - 1) - 4) + 5) - 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = (1(3cos(1π(3(2x^2 - 1x)) + 3) - 3) - 2)/(3(3cos(1π(3(2x^2 - 1x)) + 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"
}
] | 44 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(2(-2sin(2πx - 2))^2 - 2(-2sin(2πx - 2)) - 2) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -1(-1cos(1π(-2|x + 1| - 3) - 1) + 2)^2 - 3(-1cos(1π(-2|x + 1| - 3) - 1) + 2) - 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"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(2π(-5(3x^2 + 4x + 3) - 1)) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1(3sin(1πx - 3) + 2)^2 - 4(3sin(1πx - 3) + 2) - 5) + 3| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -1cos(2π(3|(2x - 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"
}
] | 77 | MATH |
[
{
"content": "For the function f(x) = 3|(-4(-4(1sin(1πx - 3) - 1) - 4)^2 + 1(-4(1sin(1πx - 3) - 1) - 4) - 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"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1cos(2π(4x) + 3) - 2) + 3)/(-3(-1cos(2π(4x) + 3) - 2)) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "For the function f(x) = -1(1(-2|(1sin(3πx + 3) - 1)| + 3))^2 - 4(1(-2|(1sin(3πx + 3) - 1)| + 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"
}
] | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-1sin(3π(-5x) - 1) - 2) - 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "For the function f(x) = (2(1cos(1π(3|(-1x^2 - 3x)| - 1) + 1) + 1) + 2)/(1(1cos(1π(3|(-1x^2 - 3x)| - 1) + 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"
}
] | 51 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(2(3cos(1πx) + 3) + 1) - 2)/(-2(2(3cos(1πx) + 3) + 1) + 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = -4(2cos(3π(2((2x + 2)/(3x - 2))) + 2) - 2)^2 + 2(2cos(3π(2((2x + 2)/(3x - 2))) + 2) - 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"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 3cos(2π(-1((-2(1x + 5) - 1)/(1(1x + 5) + 2))^2 + 1((-2(1x + 5) - 1)/(1(1x + 5) + 2)) - 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"
}
] | 12 | MATH |
[
{
"content": "For the function f(x) = -1|(5(2sin(2π(-2x + 5) + 2))^2 + 1(2sin(2π(-2x + 5) + 2)) - 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"
}
] | 78 | MATH |
[
{
"content": "For the function f(x) = 2|((1(2cos(3πx - 1) - 1) + 3)/(-3(2cos(3πx - 1) - 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"
}
] | 34 | MATH |
[
{
"content": "For the function f(x) = -1((1(-1cos(1πx - 2) - 1) - 1)/(2(-1cos(1πx - 2) - 1) + 1))^2 + 4((1(-1cos(1πx - 2) - 1) - 1)/(2(-1cos(1πx - 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}",
"ro... | 29 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3cos(2π(-1(2|x - 2| - 2) + 5) - 2) - 3)^2 + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -4(-1(-2cos(1π(-3|x + 2| + 1) - 1) - 3)^2 - 3(-2cos(1π(-3|x + 2| + 1) - 1) - 3) + 3) - 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"
}
] | 45 | MATH |
[
{
"content": "For the function f(x) = -2(4(3sin(3πx + 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"
}
] | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2π((-3(-1|x| - 2) + 3)/(-1(-1|x| - 2) - 1)) - 1) + 2 positive?\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) = -3cos(2π(1|((-1x - 2)/(-2x + 3)) - 1| + 1)) + 1 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(-2|(-5(1sin(2πx + 1) - 1)^2 - 5(1sin(2πx + 1) - 1) - 5) - 2| + 2) + 2)/(-2(-2|(-5(1sin(2πx + 1) - 1)^2 - 5(1sin(2πx + 1) - 1) - 5) - 2| + 2) + 3) is strictly increasing.\n\nPresent the answer in LaTex for... | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2π(-3((-2(-5x + 4) + 3)/(-1(-5x + 4) - 3))^2 - 5) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "For the function f(x) = -2|(-1sin(3π((-1x)/(-1x + 2)) - 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"
}
] | 13 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3π((3(1x + 1) - 1)/(-1(1x + 1) - 2)) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(2π(-3(3((-2x + 2)/(3x))^2 + 3((-2x + 2)/(3x)) + 3) + 2) - 3) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(-5((-3x - 1)/(2x + 3))^2 - 3((-3x - 1)/(2x + 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"
}
] | 14 | MATH |
[
{
"content": "For the function f(x) = (2(-2cos(3π(-3x^2 + 5x - 3) + 3)) + 2)/(2(-2cos(3π(-3x^2 + 5x - 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"
}
] | 38 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3|(3((3(-2cos(2πx + 2) - 2) + 1)/(1(-2cos(2πx + 2) - 2) + 1)) - 2) + 1| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = 2|((3(-1cos(2π(-4x^2 + 5x) + 1) + 1) + 3)/(-3(-1cos(2π(-4x^2 + 5x) + 1) + 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"
}
] | 43 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3sin(3π(1|(1x^2 - 1x - 5) - 1| + 2)) - 3))/(-2(3sin(3π(1|(1x^2 - 1x - 5) - 1| + 2)) - 3) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(1π(-2|(-3x^2 - 3x)| - 3) + 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "For the function f(x) = 5(2|(-2(1cos(3πx - 3) - 1)^2 + 1(1cos(3πx - 3) - 1) - 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"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = -4(-3|(2sin(1πx + 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"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1sin(1π(3|x + 3| + 2) + 2) + 2) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = -1(5(1cos(1πx - 1) - 3)^2 + 4(1cos(1πx - 1) - 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"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(3π(5((-1(4x - 4) - 3)/(-1(4x - 4)))^2 + 1((-1(4x - 4) - 3)/(-1(4x - 4))) - 2) + 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-2cos(3π((1(-3x^2 + 2x + 2))/(-1(-3x^2 + 2x + 2) - 2)) - 3) + 3)| + 1 is strictly increasing.\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) = 1cos(2π(-1(-2x^2 - 3x - 1) + 4) - 3) + 3 is strictly decreasing.\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) = -3(-2sin(1π(1x - 2) + 3) + 3)^2 + 5(-2sin(1π(1x - 2) + 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) = 3(-3|(2sin(1πx) + 1) - 1| - 2)^2 - 2(-3|(2sin(1πx) + 1) - 1| - 2) - 4 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) = (-2(-2cos(2π(5x^2 + 3x - 1) + 2) + 2) - 2)/(-3(-2cos(2π(5x^2 + 3x - 1) + 2) + 2) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3|(2cos(3πx + 1) - 2) - 1| + 1) + 1)/(-3(-3|(2cos(3πx + 1) - 2) - 1| + 1) - 1) 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) = 1sin(3π(4((1x - 3)/(-2x)) - 3) + 1) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-4((-1(-3cos(3πx - 3) - 3) + 3)/(1(-3cos(3πx - 3) - 3) + 3)) + 2)^2 - 4(-4((-1(-3cos(3πx - 3) - 3) + 3)/(1(-3cos(3πx - 3) - 3) + 3)) + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: ... | 12 | MATH |
[
{
"content": "For the function f(x) = (-2(3cos(1π(3x + 2) - 2) + 1) - 1)/(-2(3cos(1π(3x + 2) - 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"
}
] | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(2|(3cos(3π(1x)) + 3) - 2| - 2) + 3)/(-2(2|(3cos(3π(1x)) + 3) - 2| - 2) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "For the function f(x) = 2(3(-1sin(3πx + 2) + 2)^2 + 1(-1sin(3πx + 2) + 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 answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(2π(-1(-1|x + 1| - 1)^2 - 5(-1|x + 1| - 1) - 3) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "For the function f(x) = 1cos(1π(3(-1|x + 2| + 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"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = 2(3sin(1π(1|((-1x - 2)/(-1x + 2)) + 3| - 1) + 2) + 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"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = (3(2(-1cos(2π(2|x + 2|)) - 2)^2 + 4(-1cos(2π(2|x + 2|)) - 2) + 1) + 3)/(-2(2(-1cos(2π(2|x + 2|)) - 2)^2 + 4(-1cos(2π(2|x + 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 answe... | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3((1(3sin(3πx) - 2) - 3)/(2(3sin(3πx) - 2) + 3))^2 + 5((1(3sin(3πx) - 2) - 3)/(2(3sin(3πx) - 2) + 3)) - 3) + 3| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 49 | MATH |
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