messages listlengths 1 1 | ground_truth stringlengths 1 3 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3π(-4(3((-2x)/(-2x - 2))^2 - 3((-2x)/(-2x - 2)) - 3) + 4) - 1) + 1 is strictly increasing.\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) = -2(1(3cos(1π(3|x| + 1) + 1) + 3) + 4)^2 - 1(1(3cos(1π(3|x| + 1) + 1) + 3) + 4) + 3 is strictly increasing.\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|(-1(4((3x - 2)/(1x + 1)))^2 - 2(4((3x - 2)/(1x + 1))) + 2) + 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = (-3(3(5(3|x + 3|)^2 - 4(3|x + 3|) - 4) - 4) + 1)/(-3(3(5(3|x + 3|)^2 - 4(3|x + 3|) - 4) - 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1π(1x^2 - 5x + 5) - 1) + 3 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) = -1sin(1π(-4(1x^2 + 5x - 1) + 3)) is strictly decreasing.\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((1x - 2)/(3x + 2))^2 + 2((1x - 2)/(3x + 2)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2cos(1πx - 1) + 2) + 1| - 2 positive?\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(-1sin(2π((1x - 3)/(-3x - 1)) - 1) - 3)^2 + 3(-1sin(2π((1x - 3)/(-3x - 1)) - 1) - 3) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = -2|(1((-3(4x - 5) - 3)/(1(4x - 5) - 2))^2 - 4((-3(4x - 5) - 3)/(1(4x - 5) - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -3((2(2|x - 1| - 1) + 3)/(1(2|x - 1| - 1) + 3))^2 + 5((2(2|x - 1| - 1) + 3)/(1(2|x - 1| - 1) + 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -2(-2sin(1π(-2|x + 3| - 2) + 3) + 3)^2 - 1(-2sin(1π(-2|x + 3| - 2) + 3) + 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) = -2sin(2π(-3|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"
}
] | 61 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1((2x - 2)/(3x + 2))^2) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|((1x + 2)/(-1x - 1)) + 3| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1sin(2π((3x - 2)/(3x + 3)) - 2) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = -1|(1sin(2π(2x^2 - 2x) - 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"
}
] | 43 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(3((2x + 2)/(3x - 2))^2 - 2((2x + 2)/(3x - 2))) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 1|(1(3e^(-1x) - 2)^2 - 5(3e^(-1x) - 2) - 4) + 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = (-1(-2|(1sin(1πx + 2) + 1) + 3| - 3) - 3)/(-3(-2|(1sin(1πx + 2) + 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"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = 5(3cos(2π(5x - 1) + 1))^2 + 3(3cos(2π(5x - 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"
}
] | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(3π(-4(-3|x - 1| + 1)^2 + 5(-3|x - 1| + 1) + 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) = 4(-2cos(2πx + 1) - 1) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = (2(5(-2cos(3πx + 3) + 3)^2 + 4(-2cos(3πx + 3) + 3) + 4) - 3)/(1(5(-2cos(3πx + 3) + 3)^2 + 4(-2cos(3πx + 3) + 3) + 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": ... | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|((2(1(3cos(3πx - 1) + 2) - 5) + 3)/(3(1(3cos(3πx - 1) + 2) - 5) - 2)) + 2| positive?\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(-2cos(1πx + 3) - 3) - 1)/(2(-2cos(1πx + 3) - 3) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = 1cos(1π(-1x^2 - 3x + 4) + 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"
}
] | 55 | MATH |
[
{
"content": "For the function f(x) = (-1(-3(-1(3|x - 3| - 1) + 3)^2 - 1(-1(3|x - 3| - 1) + 3) + 2) - 2)/(-1(-3(-1(3|x - 3| - 1) + 3)^2 - 1(-1(3|x - 3| - 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}",
... | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(3π(-4(-3((2x + 3)/(1x + 1))^2 + 5((2x + 3)/(1x + 1)) + 5) + 1) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π(-2|x - 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"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = 5(-1cos(2πx - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2π(5(-1x - 3)^2 + 2(-1x - 3)) - 2) + 2 is strictly increasing.\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) = 2sin(2π(4x + 3) - 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "For the function f(x) = (3(2(3sin(2πx - 1) + 1) + 2) + 1)/(1(2(3sin(2πx - 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"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(-3(4(2|x - 2|) - 5)^2 + 4(4(2|x - 2|) - 5) + 4) + 2)/(-2(-3(4(2|x - 2|) - 5)^2 + 4(4(2|x - 2|) - 5) + 4) - 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2|(-1cos(3π(3x^2 + 3x - 1) - 1) + 1)|) + 2 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) = 1(-3|(1x - 5) + 1| + 1)^2 - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(3|(-1sin(2π(1x^2 - 5x) + 2) + 1)| + 1) - 2)/(-1(3|(-1sin(2π(1x^2 - 5x) + 2) + 1)| + 1) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "For the function f(x) = 5(-2|((-1(-1x + 3) - 1)/(3(-1x + 3) - 1)) - 1| - 3)^2 - 4(-2|((-1(-1x + 3) - 1)/(3(-1x + 3) - 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"
}... | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-2cos(3π(-2x^2 + 3x - 3) + 2) + 3) + 1 negative?\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(-1cos(2π(4x + 3) + 1) - 3)^2 + 2(-1cos(2π(4x + 3) + 1) - 3) + 1 is strictly increasing.\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(2sin(1πx + 3) - 3) - 5 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|(4(-1e^(1(-4x^2 - 3x + 2))) - 3) + 3| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2sin(1π(-1(-2x^2 + 4x + 5) + 4) - 3) + 3) + 2| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3π(5(1|(3x + 3) + 3| + 3)^2 - 3(1|(3x + 3) + 3| + 3) + 5) - 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "For the function f(x) = -2((2(-1|x + 3| + 1) - 3)/(1(-1|x + 3| + 1) + 2))^2 + 4((2(-1|x + 3| + 1) - 3)/(1(-1|x + 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"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(2cos(2π(-5((2x - 2)/(-2x - 1))^2 - 3((2x - 2)/(-2x - 1)) + 1) + 1) + 3) is strictly decreasing.\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) = -2|(-5(1((1x + 2)/(2x - 3))^2 - 5((1x + 2)/(2x - 3)) + 2) - 1) - 1| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3((3(-3|(-5x^2 - 5x + 2)| + 2) - 3)/(-3(-3|(-5x^2 - 5x + 2)| + 2) + 1)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-1cos(2πx - 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(-5(-2|(-1cos(3πx - 3) + 2) + 2|)^2 - 3(-2|(-1cos(3πx - 3) + 2) + 2|) + 5))/(-2(-5(-2|(-1cos(3πx - 3) + 2) + 2|)^2 - 3(-2|(-1cos(3πx - 3) + 2) + 2|) + 5) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex for... | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(3π((-3(-2|x + 3| + 1) - 3)/(1(-2|x + 3| + 1) - 2)) - 1) + 1 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((1(1(2|x - 2| - 2)^2 - 4(2|x - 2| - 2)) - 3)/(-3(1(2|x - 2| - 2)^2 - 4(2|x - 2| - 2)) - 3)) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 2(-3|((3(2x + 4) + 3)/(-2(2x + 4)))| - 2)^2 - 4(-3|((3(2x + 4) + 3)/(-2(2x + 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 answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(2sin(3πx + 3) - 1) + 2| - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π(-1|((-2x)/(-2x + 3)) + 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"
}
] | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1(2cos(2π(-2x^2 - 5x - 1) + 1) + 2) + 5) - 1)/(-1(1(2cos(2π(-2x^2 - 5x - 1) + 1) + 2) + 5) + 2) negative?\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(1|(-2cos(1πx - 2) + 2) + 2| - 1) + 1)/(-2(1|(-2cos(1πx - 2) + 2) + 2| - 1) - 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 2(-3|(1cos(3π((2x + 3)/(-3x - 3)) - 1) - 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"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-1(-3(-2e^(1x) + 3) - 3)^2 + 1(-3(-2e^(1x) + 3) - 3) + 2) + 1| - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -2|(3sin(1π(2(-1x^2 - 5x - 3) - 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"
}
] | 53 | MATH |
[
{
"content": "For the function f(x) = (1(-3|(-3(-1x^2 - 3x - 2) - 3) + 3| + 3) + 3)/(2(-3|(-3(-1x^2 - 3x - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -2sin(2π(-5x - 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"
}
] | 100 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3sin(3π(5(-2(-3|x| + 2) - 1)^2 + 5(-2(-3|x| + 2) - 1) + 1) + 2) - 1 is strictly increasing.\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) = (-1(3sin(1πx - 2) + 2) - 1)/(1(3sin(1πx - 2) + 2)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -1sin(3π((3x + 1)/(-1x))) - 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"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2π(1|(5x - 1)| - 3)) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 90 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π(4(1((-2x - 1)/(2x - 1)))^2 - 4(1((-2x - 1)/(2x - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(1π(-1|x + 3| - 2)) is strictly increasing.\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) = -4(-3|(-2cos(1πx + 2) + 2) - 3| + 1)^2 + 4(-3|(-2cos(1πx + 2) + 2) - 3| + 1) - 2 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) = -3(-3sin(1π(5(2|x + 2| + 2)^2 - 4(2|x + 2| + 2)) + 1) + 1) + 3 negative?\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(-3(2|(-2x) + 2| - 1)^2 - 3(2|(-2x) + 2| - 1) - 1) - 1)/(-3(-3(2|(-2x) + 2| - 1)^2 - 3(2|(-2x) + 2| - 1) - 1) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 2 | MATH |
[
{
"content": "For the function f(x) = (-2(1|(3sin(3π(4x) - 1) + 3) + 1| + 2) + 1)/(-2(1|(3sin(3π(4x) - 1) + 3) + 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"
}
] | 80 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(4(2sin(3π((-1x)/(-1x - 1)) + 3)) + 5)^2 + 1(4(2sin(3π((-1x)/(-1x - 1)) + 3)) + 5) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = 1|(5((-3x - 3)/(3x - 2)) + 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"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(2cos(3π(2|x - 1| + 3) - 3) + 2) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(4(1cos(1π((1x + 1)/(2x - 1)) - 2)) + 4)^2 - 4(4(1cos(1π((1x + 1)/(2x - 1)) - 2)) + 4) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = 3cos(1π(-5(3x^2 - 1x - 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"
}
] | 35 | MATH |
[
{
"content": "For the function f(x) = -2sin(3π(-5x^2 - 5x - 4) - 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"
}
] | 34 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π(4(1|x| - 3)^2 - 2(1|x| - 3) - 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"
}
] | 54 | MATH |
[
{
"content": "For the function f(x) = 1sin(1π((-2x)/(2x - 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"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = (1(-2|(1(-2x^2 + 4x + 1) + 2) - 3| - 1))/(-1(-2|(1(-2x^2 + 4x + 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 4(2cos(1π((1x - 2)/(-2x - 3)) - 1) - 1)^2 + 3(2cos(1π((1x - 2)/(-2x - 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"
}
] | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-1((-3x - 3)/(-3x + 2))^2 - 4((-3x - 3)/(-3x + 2)) + 2) - 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(1π(1|x - 2| - 3) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2π(-1x - 4) - 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -3e^(2(-2(3(-1|x - 2| + 2) + 4)^2 + 3(3(-1|x - 2| + 2) + 4) - 5)) + 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"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = 2|((3x - 3)/(2x + 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3e^(-1(-3(-1|x - 3| + 3)^2 + 1(-1|x - 3| + 3) + 4)) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 4((-1x + 3)/(1x))^2 - 4((-1x + 3)/(1x)) - 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2sin(1πx + 3))^2 - 1(-2sin(1πx + 3)) - 5 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) = (2(-3sin(1π(-1|x + 3| + 3) - 3) - 3) - 3)/(1(-3sin(1π(-1|x + 3| + 3) - 3) - 3) + 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -1(-3sin(1πx) + 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"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 4(-1sin(1πx) + 3) + 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 1|(5((1(-2x + 4) + 2)/(2(-2x + 4)))^2 + 4((1(-2x + 4) + 2)/(2(-2x + 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 answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(4(-3cos(3πx + 1) + 1) - 1))/(-1(4(-3cos(3πx + 1) + 1) - 1) - 3) is strictly increasing.\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) = 5((-1(5(-1|x - 1|) + 4) + 1)/(-2(5(-1|x - 1|) + 4) + 3))^2 + 1((-1(5(-1|x - 1|) + 4) + 1)/(-2(5(-1|x - 1|) + 4) + 3)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 2 | MATH |
[
{
"content": "For the function f(x) = -2((-3(-1cos(1πx + 3) + 1) + 1)/(3(-1cos(1πx + 3) + 1) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2π(-2x + 2)) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -1((2(4(-1|x + 1| + 1) + 3) - 3)/(2(4(-1|x + 1| + 1) + 3) + 2))^2 + 2((2(4(-1|x + 1| + 1) + 3) - 3)/(2(4(-1|x + 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}",... | 2 | MATH |
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