messages list | ground_truth string | dataset string |
|---|---|---|
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1((-1(1cos(3π(2|x + 2| + 2) - 1) - 3) + 2)/(-1(1cos(3π(2|x + 2| + 2) - 1) - 3))) + 2 negative?\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) = -1cos(2π(2|(3((1x)/(-2x - 2)) + 2) + 3| + 1) + 1) - 3 positive?\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) = 4(-2(2sin(3πx) - 1) - 1)^2 + 1(-2(2sin(3πx) - 1) - 1) is strictly increasing.\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|(-2sin(3π((1x - 1)/(-1x + 2)) + 1) - 3) + 3| + 3 negative?\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) = -3sin(2π((1(1|x + 2| - 3) + 3)/(-1(1|x + 2| - 3) + 2)) + 1) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "For the function f(x) = (-3(-3cos(3π(-2|(3x + 1) - 3| + 2) - 1) + 2) - 1)/(-3(-3cos(3π(-2|(3x + 1) - 3| + 2) - 1) + 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"
}
] | 18 | MATH |
[
{
"content": "For the function f(x) = -1((3(1cos(3πx + 3) - 2) - 3)/(-3(1cos(3πx + 3) - 2) - 3))^2 + 3((3(1cos(3πx + 3) - 2) - 3)/(-3(1cos(3πx + 3) - 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... | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(3sin(2π(-4(-3|x + 2| - 2)^2 - 1(-3|x + 2| - 2) - 1) - 1) + 2) - 3)/(3(3sin(2π(-4(-3|x + 2| - 2)^2 - 1(-3|x + 2| - 2) - 1) - 1) + 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 24 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-2cos(2π(-3|x - 2| + 3) + 2) - 2) + 2)/(1(-2cos(2π(-3|x - 2| + 3) + 2) - 2) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "For the function f(x) = -1cos(1π(-2((-2(-1|x + 2| + 3))/(2(-1|x + 2| + 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: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-1sin(1π((-2x + 2)/(1x))) - 1)^2 - 5(-1sin(1π((-2x + 2)/(1x))) - 1) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = 1sin(3π(2|(2(-1x^2 - 3x + 4) - 4) - 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"
}
] | 49 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(1cos(3π((-1(3x^2 - 1x + 3) - 3)/(1(3x^2 - 1x + 3) - 3)) - 1) + 2) - 2| + 2 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) = -2cos(1π(1|(5(3x^2 + 3) + 2) + 2|) - 2) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "For the function f(x) = -5(-1(-2|(1sin(2πx + 2) + 3) + 2| + 2) + 4)^2 - 1(-1(-2|(1sin(2πx + 2) + 3) + 2| + 2) + 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"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2π(2(-1|(1x^2 + 5x - 4) + 1| - 2) + 3) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(2(1cos(2π(4x + 4) - 1) + 1)^2 + 1(1cos(2π(4x + 4) - 1) + 1)) - 2)/(-2(2(1cos(2π(4x + 4) - 1) + 1)^2 + 1(1cos(2π(4x + 4) - 1) + 1)) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\box... | 81 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(3π(1((3x - 2)/(2x - 3))^2 - 2((3x - 2)/(2x - 3)) - 2) - 1) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = -3|(2cos(3π(-4((-1x - 2)/(-2x - 2))^2 + 4((-1x - 2)/(-2x - 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"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = -1sin(1π(-2|((2(-3x^2 - 3x + 5) - 2)/(1(-3x^2 - 3x + 5) + 1)) + 3| + 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"
}
] | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-3|((-3(-3cos(2πx - 2) + 1) + 3)/(-1(-3cos(2πx - 2) + 1) + 2))| - 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(3cos(3π((-2(-4x^2 + 4x + 1) + 3)/(3(-4x^2 + 4x + 1) - 1)) + 1)) - 2| - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = (-1(-4(1cos(2π(-1|x| + 2) + 2) + 2)^2 + 5(1cos(2π(-1|x| + 2) + 2) + 2) + 1) + 2)/(-3(-4(1cos(2π(-1|x| + 2) + 2) + 2)^2 + 5(1cos(2π(-1|x| + 2) + 2) + 2) + 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex for... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(2cos(2π((2(1x - 1) + 3)/(1(1x - 1) - 3)) - 1) + 2)^2 - 5(2cos(2π((2(1x - 1) + 3)/(1(1x - 1) - 3)) - 1) + 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(4(-3|x + 2| - 1)^2 - 5(-3|x + 2| - 1) - 5) - 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"
}
] | 18 | MATH |
[
{
"content": "For the function f(x) = (3(-1|(1sin(2πx) - 1) - 2| + 3) - 1)/(-2(-1|(1sin(2πx) - 1) - 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"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-4((-1(-1sin(2πx + 3) - 2) + 1)/(-1(-1sin(2πx + 3) - 2) - 2))^2 + 1((-1(-1sin(2πx + 3) - 2) + 1)/(-1(-1sin(2πx + 3) - 2) - 2)) - 2) - 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 40 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3π(-1|(3x^2 + 1x - 5)|) - 3) is strictly increasing.\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) = (-3(1|(1cos(2πx + 3)) - 2| - 3) - 1)/(-1(1|(1cos(2πx + 3)) - 2| - 3) - 1) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -1cos(3π(-3((2x - 2)/(-1x + 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"
}
] | 32 | MATH |
[
{
"content": "For the function f(x) = (-3(3(2sin(1πx + 2) + 1) + 4) + 1)/(-1(3(2sin(1πx + 2) + 1) + 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 answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(2(-1sin(1πx - 2) - 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) = (-1(-4(1|(3sin(1πx - 3) - 2) + 3| - 2)^2 - 3(1|(3sin(1πx - 3) - 2) + 3| - 2) - 3) + 3)/(2(-4(1|(3sin(1πx - 3) - 2) + 3| - 2)^2 - 3(1|(3sin(1πx - 3) - 2) + 3| - 2) - 3) + 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer i... | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1sin(2π(-3(-1|x - 2| - 2) + 2) - 3) + 1)^2 - 1(1sin(2π(-3(-1|x - 2| - 2) + 2) - 3) + 1) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 65 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1cos(2π((-1(-1|x + 3| - 1) - 1)/(-1(-1|x + 3| - 1) - 3)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-3cos(1π(-2x) + 3) - 3) - 1)/(3(-3cos(1π(-2x) + 3) - 3) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -2cos(3π(-3((-1x + 3)/(2x + 1)) - 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"
}
] | 18 | MATH |
[
{
"content": "For the function f(x) = 2(-2cos(3π((-3(-1|x| + 2) + 2)/(-3(-1|x| + 2))) + 3) + 2)^2 + 1(-2cos(3π((-3(-1|x| + 2) + 2)/(-3(-1|x| + 2))) + 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"... | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-4(-2(3sin(2πx + 2) - 2)^2 - 4(3sin(2πx + 2) - 2) + 4) + 1) - 3| + 2 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) = -2|(-2cos(2π((1x + 3)/(3x + 3)) - 1)) - 3| + 3 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(1sin(3π(-2|x + 3| + 1) + 2) + 3)^2 - 4(1sin(3π(-2|x + 3| + 1) + 2) + 3) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "For the function f(x) = -3sin(1π(-2|(-3((2x + 3)/(2x))^2 + 3((2x + 3)/(2x)) - 1) - 2| + 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"
}
] | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(1π(-4((-2(-2x - 3) + 1)/(-1(-2x - 3) - 2))^2 + 4((-2(-2x - 3) + 1)/(-1(-2x - 3) - 2)) - 4) + 2) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(3|(3sin(2πx) + 1)| + 3) - 2)/(3(3|(3sin(2πx) + 1)| + 3) + 2) positive?\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(-2sin(3π((2(-4x^2 - 2x + 2) - 3)/(-1(-4x^2 - 2x + 2) + 3)) + 3) + 1) - 3 is strictly decreasing.\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) = (2(1(1cos(2πx - 3) - 2) - 2) + 1)/(2(1(1cos(2πx - 3) - 2) - 2) + 2) is strictly decreasing.\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((1(-2sin(1πx + 2) - 3) - 2)/(3(-2sin(1πx + 2) - 3) - 1))^2 + 5((1(-2sin(1πx + 2) - 3) - 2)/(3(-2sin(1πx + 2) - 3) - 1)) - 1) + 2| + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\b... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|(-1sin(1π(-5(-5x + 3)^2 - 1(-5x + 3) - 3) - 2) - 1) + 3| + 3 positive?\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|((-2(-3sin(3π(3x - 4) + 3) - 2) + 1)/(1(-3sin(3π(3x - 4) + 3) - 2))) + 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "For the function f(x) = 2sin(3π(-1|(3((1x - 3)/(-1x)) + 2) + 3| + 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"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = 2|((3(-3cos(1πx - 1) + 3) + 2)/(1(-3cos(1πx - 1) + 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"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = 1cos(3π(1(3(-3|x - 3| - 2)^2 - 3(-3|x - 3| - 2) - 5) + 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"
}
] | 17 | MATH |
[
{
"content": "For the function f(x) = (3(5(-1sin(1πx - 2) - 1)^2 - 2(-1sin(1πx - 2) - 1) - 4) + 2)/(1(5(-1sin(1πx - 2) - 1)^2 - 2(-1sin(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": ... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((3(2sin(2πx) + 2) - 2)/(-3(2sin(2πx) + 2) - 3))^2 + 2((3(2sin(2πx) + 2) - 2)/(-3(2sin(2πx) + 2) - 3)) - 3 positive?\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(1sin(1πx + 2) + 3)^2 - 1(1sin(1πx + 2) + 3) + 5) + 2| + 3 positive?\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) = (-2(-3(2|(2sin(3πx - 2) - 3) - 1| - 2)^2 - 2(2|(2sin(3πx - 2) - 3) - 1| - 2) - 3) - 3)/(1(-3(2|(2sin(3πx - 2) - 3) - 1| - 2)^2 - 2(2|(2sin(3πx - 2) - 3) - 1| - 2) - 3) - 3) negative?\n\nPresent the answer in LaTex... | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2|((3(2cos(3πx + 1) - 3) + 2)/(3(2cos(3πx + 1) - 3))) - 1| + 1) negative?\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) = -3sin(1π(4(-1((-1x + 2)/(1x + 2))^2 - 5((-1x + 2)/(1x + 2))) + 5) + 2) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(3π(5(-4x^2 + 5x - 5) - 1) - 2) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -1|((1(-5(1cos(1πx + 2) - 1) - 1) + 3)/(1(-5(1cos(1πx + 2) - 1) - 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{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -1(1(-1cos(3π(3|x + 1| - 3) - 1) - 2) + 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"
}
] | 90 | MATH |
[
{
"content": "For the function f(x) = 2|(-2cos(2π(-3x + 5) + 3) - 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 answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "For the function f(x) = 1(1|(4(1cos(2πx + 1) - 1) - 2) + 2| + 3)^2 - 3(1|(4(1cos(2πx + 1) - 1) - 2) + 2| + 3) + 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": "For the function f(x) = (-1(-2cos(3π(2x^2 + 2x - 2) + 1)) - 3)/(-2(-2cos(3π(2x^2 + 2x - 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"
}
] | 47 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π(1|(2(-1x + 3)^2 + 3(-1x + 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"
}
] | 60 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(2sin(3π(1|(1x^2 + 4x) + 1| - 3) + 3) - 3) - 2)/(2(2sin(3π(1|(1x^2 + 4x) + 1| - 3) + 3) - 3) + 3) negative?\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(1π(-3|(4x + 2) - 3|) + 3) + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "For the function f(x) = -2(4(-2cos(3πx + 2) + 1)^2 + 3(-2cos(3πx + 2) + 1) + 4) - 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"
}
] | 53 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3π(-2|(3x - 3) + 1| + 3) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = 3sin(3π(1|(-5x^2 - 4x + 4) - 3| + 1) - 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"
}
] | 32 | MATH |
[
{
"content": "For the function f(x) = (1(-3sin(1π(-3x^2 + 2x)) - 1) + 3)/(1(-3sin(1π(-3x^2 + 2x)) - 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"
}
] | 39 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(1π(2((-3(-1|x + 2| - 2) - 2)/(-1(-1|x + 2| - 2) + 2))^2 - 1((-3(-1|x + 2| - 2) - 2)/(-1(-1|x + 2| - 2) + 2)) - 2) + 1) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2π(1(3|x + 1| + 1) + 4) - 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "For the function f(x) = (-1(-3sin(2π(4(1x + 3)^2 + 4(1x + 3) - 5) - 3) + 2) - 2)/(2(-3sin(2π(4(1x + 3)^2 + 4(1x + 3) - 5) - 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"
... | 48 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((-1(2sin(3πx - 3) - 1) - 3)/(-1(2sin(3πx - 3) - 1) + 2)) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = (-1(3(-1sin(1π(-1x + 4) - 1) - 2)^2 - 4(-1sin(1π(-1x + 4) - 1) - 2) + 1) - 2)/(-3(3(-1sin(1π(-1x + 4) - 1) - 2)^2 - 4(-1sin(1π(-1x + 4) - 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: \... | 10 | MATH |
[
{
"content": "For the function f(x) = -2((3(1|(2sin(3πx - 2)) + 3| - 1) - 2)/(1(1|(2sin(3πx - 2)) + 3| - 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"
}
] | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|(4(2sin(2πx + 1) + 2)^2 + 4(2sin(2πx + 1) + 2) + 5) + 2| - 1) - 1 is strictly decreasing.\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(-1cos(1π(-4(1|x - 3|) - 3) - 2))^2 + 5(-1cos(1π(-4(1|x - 3|) - 3) - 2)) is strictly increasing.\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) = 5((-2(2|(-3sin(1πx - 3) + 3) - 3|) - 2)/(2(2|(-3sin(1πx - 3) + 3) - 3|) + 3)) - 2 positive?\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) = 3sin(1π(-4((-2x - 3)/(-3x + 1))^2 + 3((-2x - 3)/(-3x + 1)) + 2)) - 1 is strictly decreasing.\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) = (1(1(-1cos(3πx + 2) + 1) + 1) + 3)/(1(1(-1cos(3πx + 2) + 1) + 1) + 1) 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) = (-2(-5(-2(2sin(2πx - 3) + 1) + 3)^2 - 2(-2(2sin(2πx - 3) + 1) + 3) + 1) + 2)/(-1(-5(-2(2sin(2πx - 3) + 1) + 3)^2 - 2(-2(2sin(2πx - 3) + 1) + 3) + 1) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3(-1sin(2π(-3x^2 + 3x - 4)) - 2) - 4) + 1)/(-2(-3(-1sin(2π(-3x^2 + 3x - 4)) - 2) - 4) + 3) is strictly increasing.\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) = (-1(-3cos(1π(1|(2x^2 - 3x + 4) + 1| - 2)) + 2) - 3)/(3(-3cos(1π(1|(2x^2 - 3x + 4) + 1| - 2)) + 2) - 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π(3((-2x - 1)/(3x + 3))^2 - 3((-2x - 1)/(3x + 3)) - 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"
}
] | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(1π(-1|(-2(1x - 5)^2 + 2(1x - 5) + 5) + 2| + 3) - 2) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(1π(4(-3|(1x + 5) + 3| + 1)^2 + 2(-3|(1x + 5) + 3| + 1) + 4)) - 3 positive?\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(-2(1cos(1πx) - 1) - 1)^2 - 1(-2(1cos(1πx) - 1) - 1) - 2) - 3| - 3 is strictly decreasing.\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) = 1sin(1π(-5(-4x^2 + 1)) - 2) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "For the function f(x) = 1|((2(3(1cos(3πx) + 2)^2 - 2(1cos(3πx) + 2) + 4) - 1)/(1(3(1cos(3πx) + 2)^2 - 2(1cos(3πx) + 2) + 4) - 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) = -2cos(3π(-3(-1(3|x + 2| + 1)^2 - 5(3|x + 2| + 1) + 3) - 2)) + 3 negative?\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(-2sin(2π(-2|((-1x + 1)/(-3x))| - 1) - 2) + 1) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "For the function f(x) = (-2(-5(-3sin(3π(1|x - 3| + 1) - 3) + 3)) - 2)/(1(-5(-3sin(3π(1|x - 3| + 1) - 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 answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3(2cos(1π(-1x - 4) - 1) - 1) + 3)/(3(2cos(1π(-1x - 4) - 1) - 1)) is strictly decreasing.\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) = 3|((1(5(2cos(1πx)) + 4) + 3)/(2(5(2cos(1πx)) + 4) - 3)) - 2| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "For the function f(x) = -5((1(-2sin(2πx - 2) - 2) - 3)/(3(-2sin(2πx - 2) - 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"
}
] | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2π(-4(2|x + 3| - 3) + 3) + 3) - 1 is strictly increasing.\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) = 1|((2(-1cos(2π(5x^2 + 1x - 5) + 1)) + 3)/(2(-1cos(2π(5x^2 + 1x - 5) + 1)) + 2)) - 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(3|(-4x - 4) - 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"
}
] | 76 | MATH |
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