messages listlengths 1 1 | ground_truth stringclasses 87
values | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(4(-1|(-3cos(3πx + 3) + 2) + 2| + 1)^2 - 4(-1|(-3cos(3πx + 3) + 2) + 2| + 1) - 3)^2 + 5(4(-1|(-3cos(3πx + 3) + 2) + 2| + 1)^2 - 4(-1|(-3cos(3πx + 3) + 2) + 2| + 1) - 3) + 2 positive?\n\nPresent the answer in LaTe... | 13 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2π(-3|(-3(2x)^2 - 1(2x) - 3) + 1| + 1) - 3) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "For the function f(x) = (2(2cos(1π(1(-4x - 2) + 2) - 1)) - 3)/(-3(2cos(1π(1(-4x - 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"
}
] | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1cos(3π((1(2x - 3) - 3)/(3(2x - 3) + 2)) + 2) - 3) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 5(-2(1sin(2π(1x^2 + 3x - 3) - 2) - 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"
}
] | 47 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2π((3(-3|(4x + 1)| + 2) + 1)/(1(-3|(4x + 1)| + 2) - 3)) + 1) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(2(-2|(2x + 3) + 2| + 3)^2 - 4(-2|(2x + 3) + 2| + 3))^2 + 1(2(-2|(2x + 3) + 2| + 3)^2 - 4(-2|(2x + 3) + 2| + 3)) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2|(3cos(1π(-1|x + 3| + 2) + 1) - 3) - 1| + 3) + 5 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) = -2cos(3π(-3((2((-2x + 2)/(2x - 1)))/(3((-2x + 2)/(2x - 1)) - 2))^2 - 1((2((-2x + 2)/(2x - 1)))/(3((-2x + 2)/(2x - 1)) - 2)) - 5) + 1) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5 | MATH |
[
{
"content": "For the function f(x) = 2|(-3sin(2π(-1(-4x^2 - 3x - 1)^2 + 2(-4x^2 - 3x - 1) - 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"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = -3|(1cos(2π(1|(1|x| - 1) - 3| + 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"
}
] | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-3cos(2π(3|((2x - 1)/(-1x - 2)) + 1| - 3) + 2) - 3) + 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "For the function f(x) = -5(1sin(2π(1((2x - 1)/(-1x - 1)) - 4)) + 1)^2 + 1(1sin(2π(1((2x - 1)/(-1x - 1)) - 4)) + 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"
}
] | 14 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1π((2((-3(-5x^2 - 5x - 3) - 1)/(3(-5x^2 - 5x - 3) + 2)))/(-2((-3(-5x^2 - 5x - 3) - 1)/(3(-5x^2 - 5x - 3) + 2)) - 2)) + 2) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your... | 29 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(4(1|(2cos(2πx + 2) - 3) - 1| - 2) + 2)| - 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) = 2|(-1cos(2π((1(1x^2 + 3x + 1) - 1)/(-3(1x^2 + 3x + 1))) - 3) - 3)| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = 2sin(1π(2|((-3(-2x^2 - 4x + 3) - 1)/(-2(-2x^2 - 4x + 3) + 1)) - 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"
}
] | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2((2((-3(2sin(3πx - 1) + 2) - 3)/(-2(2sin(3πx - 1) + 2) + 1)) + 1)/(1((-3(2sin(3πx - 1) + 2) - 3)/(-2(2sin(3πx - 1) + 2) + 1)) + 3))^2 - 3((2((-3(2sin(3πx - 1) + 2) - 3)/(-2(2sin(3πx - 1) + 2) + 1)) + 1)/(1((-3(2... | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((1(2cos(2π((-3x + 3)/(-3x + 1)) + 3) - 2) + 1)/(-2(2cos(2π((-3x + 3)/(-3x + 1)) + 3) - 2) + 3))^2 - 4((1(2cos(2π((-3x + 3)/(-3x + 1)) + 3) - 2) + 1)/(-2(2cos(2π((-3x + 3)/(-3x + 1)) + 3) - 2) + 3)) - 4 is... | 6 | MATH |
[
{
"content": "For the function f(x) = -1(-5(2cos(2π((-1x + 2)/(-1x - 1)) - 1) - 2)^2 + 4(2cos(2π((-1x + 2)/(-1x - 1)) - 1) - 2) + 3)^2 + 1(-5(2cos(2π((-1x + 2)/(-1x - 1)) - 1) - 2)^2 + 4(2cos(2π((-1x + 2)/(-1x - 1)) - 1) - 2) + 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.... | 6 | MATH |
[
{
"content": "For the function f(x) = -1|(1|(-2sin(3π(2x + 5) - 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"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -1cos(3π(-5(2((3x + 1)/(-3x - 3)) - 1) + 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"
}
] | 18 | MATH |
[
{
"content": "For the function f(x) = (-3(-3|(2cos(1π((-3x + 2)/(-1x - 2))) + 1) + 1| - 2))/(-1(-3|(2cos(1π((-3x + 2)/(-1x - 2))) + 1) + 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"
}... | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1(1|(1|x + 1| - 2)| + 2)^2 + 5(1|(1|x + 1| - 2)| + 2) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = 1(-2cos(3π((1(4x^2 + 5x + 2) - 2)/(2(4x^2 + 5x + 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": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(3(4(3cos(1πx - 3) + 2) + 5) - 5) + 2| + 1 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(2|(-5(-1x^2 + 5x - 1)^2 + 2(-1x^2 + 5x - 1) + 4) + 2| - 3) - 3)/(-2(2|(-5(-1x^2 + 5x - 1)^2 + 2(-1x^2 + 5x - 1) + 4) + 2| - 3) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (1(-1sin(1π((-2(-5x^2 + 3x + 2) - 3)/(-1(-5x^2 + 3x + 2))) + 3) + 3) - 1)/(3(-1sin(1π((-2(-5x^2 + 3x + 2) - 3)/(-1(-5x^2 + 3x + 2))) + 3) + 3) + 1) is strictly decreasing.\n\nPresent the answer in LaTex form... | 7 | MATH |
[
{
"content": "For the function f(x) = 3(1|(2cos(3π(-2|x + 2| + 3) - 2) - 2) - 2| + 2)^2 - 2(1|(2cos(3π(-2|x + 2| + 3) - 2) - 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"
}
] | 19 | MATH |
[
{
"content": "For the function f(x) = -1(1(-1cos(3π(-3x^2 + 5x - 3) + 2) + 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"
}
] | 50 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1|((2(2sin(2πx - 2) - 2) - 2)/(-1(2sin(2πx - 2) - 2) - 3)) + 3| + 2)^2 - 4(1|((2(2sin(2πx - 2) - 2) - 2)/(-1(2sin(2πx - 2) - 2) - 3)) + 3| + 2) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 19 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2((-1(2cos(3π(-1|x + 2| + 2) + 2) + 1) + 3)/(-2(2cos(3π(-1|x + 2| + 2) + 2) + 1) + 1))^2 + 3((-1(2cos(3π(-1|x + 2| + 2) + 2) + 1) + 3)/(-2(2cos(3π(-1|x + 2| + 2) + 2) + 1) + 1)) is strictly decreasing.\n\nPr... | 51 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((-3(-3(3sin(1πx + 1) - 3)^2 + 3(3sin(1πx + 1) - 3) - 1) - 3)/(-1(-3(3sin(1πx + 1) - 3)^2 + 3(3sin(1πx + 1) - 3) - 1) - 3))^2 + 3((-3(-3(3sin(1πx + 1) - 3)^2 + 3(3sin(1πx + 1) - 3) - 1) - 3)/(-1(-3(3sin(1πx + 1) ... | 14 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(3|((-2(2cos(2πx + 3) + 3) - 3)/(-3(2cos(2πx + 3) + 3) - 3)) - 3| + 3) - 1| + 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) = 5(1(-2cos(2π(1x^2 + 4x + 5) - 2) + 1) + 2)^2 - 4(1(-2cos(2π(1x^2 + 4x + 5) - 2) + 1) + 2) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(-1(3(-1sin(1πx - 2) - 2) - 2) + 3) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = (3(4(-2|(2sin(2πx - 3) - 3) + 1| - 1)^2 + 1(-2|(2sin(2πx - 3) - 3) + 1| - 1) - 1) - 3)/(-3(4(-2|(2sin(2πx - 3) - 3) + 1| - 1)^2 + 1(-2|(2sin(2πx - 3) - 3) + 1| - 1) - 1) - 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer... | 21 | MATH |
[
{
"content": "For the function f(x) = -3|(3(-2cos(3π((-2x - 2)/(-1x + 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"
}
] | 16 | MATH |
[
{
"content": "For the function f(x) = (2(-4((2(3sin(2πx + 3) + 2) + 3)/(-3(3sin(2πx + 3) + 2) + 4))^2 + 3((2(3sin(2πx + 3) + 2) + 3)/(-3(3sin(2πx + 3) + 2) + 4)) + 1) + 2)/(-1(-4((2(3sin(2πx + 3) + 2) + 3)/(-3(3sin(2πx + 3) + 2) + 4))^2 + 3((2(3sin(2πx + 3) + 2) + 3)/(-3(3sin(2πx + 3) + 2) + 4)) + 1) + 3), find... | 59 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(2π(1(3|(1x + 5) - 1| - 2)^2 - 5(3|(1x + 5) - 1| - 2) + 4) + 1) + 1 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) = 3|(-1|(-5(-3sin(3πx + 2) - 2) - 1) + 3| + 2) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "For the function f(x) = (-2(-1|((1(-1sin(2πx + 2) - 1) + 1)/(-3(-1sin(2πx + 2) - 1) - 2)) + 3| - 1) + 1)/(-3(-1|((1(-1sin(2πx + 2) - 1) + 1)/(-3(-1sin(2πx + 2) - 1) - 2)) + 3| - 1) - 3), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex for... | 24 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(1(-4(3cos(2πx - 1) - 1) - 4) + 1) - 3 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) = -5(2sin(1π(3(-3|x| - 1) + 2) - 2) + 3) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 91 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-4((1(1sin(1πx) + 2) - 2)/(2(1sin(1πx) + 2) - 1))) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π((-2((3(-5x + 2) + 3)/(-3(-5x + 2) - 1)) - 1)/(1((3(-5x + 2) + 3)/(-3(-5x + 2) - 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"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3|(1|(-1cos(2πx + 2) + 2) - 2| + 3) - 2| + 2) + 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = 5(5(1|(2|x + 2| - 3) - 1|)^2 + 4(1|(2|x + 2| - 3) - 1|) - 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"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-4(-1|(1cos(2πx + 1)) + 3| - 2)^2 - 1(-1|(1cos(2πx + 1)) + 3| - 2) + 1) + 2| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -1(2|(3cos(2π(1x^2 - 3x + 2) + 1)) + 2| - 3)^2 - 1(2|(3cos(2π(1x^2 - 3x + 2) + 1)) + 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"
}
] | 24 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(2π(5(3(-1|x - 3| + 3) - 4)^2 - 5(3(-1|x - 3| + 3) - 4) + 3) + 1) - 2 is strictly increasing.\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) = 3|(-4((2(1cos(3πx + 3) + 2) - 3)/(-3(1cos(3πx + 3) + 2)))^2 + 1((2(1cos(3πx + 3) + 2) - 3)/(-3(1cos(3πx + 3) + 2))) - 3) + 3| - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 43 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3(4(1cos(2πx + 1) - 1) + 5) - 3)^2 + 3(-3(4(1cos(2πx + 1) - 1) + 5) - 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = -1cos(1π(-3|(3(-2x^2 - 2x - 4)^2 + 2(-2x^2 - 2x - 4)) + 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"
}
] | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 5(1(1cos(3π(-3x + 3) + 3)) + 4) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 91 | MATH |
[
{
"content": "For the function f(x) = 3|(-3sin(1π(-5(-3x^2 - 3x + 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"
}
] | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(1cos(2π(1|(-2x^2 + 2x) + 2| - 3) - 1) - 3) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2((1((3(-2cos(3πx + 2) + 3) - 3)/(-1(-2cos(3πx + 2) + 3) - 1)) - 1)/(-3((3(-2cos(3πx + 2) + 3) - 3)/(-1(-2cos(3πx + 2) + 3) - 1)) - 1))^2 + 4((1((3(-2cos(3πx + 2) + 3) - 3)/(-1(-2cos(3πx + 2) + 3) - 1)) - 1)/(-3((... | 35 | MATH |
[
{
"content": "For the function f(x) = (2(-5(2sin(1π(-3x^2 + 4) + 3) - 2) - 2) - 1)/(-2(-5(2sin(1π(-3x^2 + 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"
}
] | 37 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-1(2|(2cos(3πx - 3) - 1) + 3| + 2) - 3)^2 + 2(-1(2|(2cos(3πx - 3) - 1) + 3| + 2) - 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = 1(3|(-1cos(1π(-3x + 5) - 3) - 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"
}
] | 31 | MATH |
[
{
"content": "For the function f(x) = (-2(5(1(1sin(1πx + 1) + 1) - 4) - 2))/(-2(5(1(1sin(1πx + 1) + 1) - 4) - 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"
}
] | 11 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((3(-3sin(3π((-2x - 1)/(2x + 2)) + 3)) + 3)/(-3(-3sin(3π((-2x - 1)/(2x + 2)) + 3)) - 2))^2 + 4((3(-3sin(3π((-2x - 1)/(2x + 2)) + 3)) + 3)/(-3(-3sin(3π((-2x - 1)/(2x + 2)) + 3)) - 2)) + 3 is strictly decrea... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1cos(2π(4((3x)/(1x - 3)) + 5)) - 1) - 1)/(3(-1cos(2π(4((3x)/(1x - 3)) + 5)) - 1) + 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "For the function f(x) = (1(2cos(3π(1(5x^2 + 1x - 1) - 5) - 3) + 2) - 3)/(-2(2cos(3π(1(5x^2 + 1x - 1) - 5) - 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"
}
] | 22 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(2(-2(1|x - 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"
}
] | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3π((-2(4((1x)/(-1x + 2)) + 2) + 3)/(-2(4((1x)/(-1x + 2)) + 2))) - 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -3cos(3π(1|((2(5x + 5) + 3)/(-2(5x + 5) + 3)) - 2| + 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"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(2π(-2(3((-1x)/(-2x + 3)) - 2)^2 + 5) + 2) - 3 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) = -1|(2|(1|(2x - 2) + 1| - 2) - 2| - 1) - 1| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3(-2sin(1π(2|(2|x - 1| - 1) + 1|) - 3) + 3)^2 + 2 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) = 3cos(3π(1(3|(1x + 1) + 3| - 1)^2 - 4(3|(1x + 1) + 3| - 1) + 1) - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "For the function f(x) = -3sin(2π(-1(1(-1x^2 + 1x + 5) + 4)^2 - 5(1(-1x^2 + 1x + 5) + 4)) - 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(2sin(2π(-2(2x - 3)^2 + 1(2x - 3) - 3)) + 1) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3((2(-2cos(1π((2x + 2)/(1x + 2))) - 2) - 1)/(3(-2cos(1π((2x + 2)/(1x + 2))) - 2) - 1)) + 2)/(-3((2(-2cos(1π((2x + 2)/(1x + 2))) - 2) - 1)/(3(-2cos(1π((2x + 2)/(1x + 2))) - 2) - 1)) + 1) is strictly decreas... | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3sin(3π(5(-2|x - 1|) - 1)) + 1)^2 + 4(-3sin(3π(5(-2|x - 1|) - 1)) + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "For the function f(x) = 2|(4(2|(1x^2 + 5x + 1) + 3| - 2)^2 - 3(2|(1x^2 + 5x + 1) + 3| - 2) - 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"
}
] | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3|(-1(-1cos(2πx + 3))^2 - 3(-1cos(2πx + 3))) + 3| - 2) + 1| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = -5(-1|(-1(-3sin(3πx - 2) + 2) + 2) - 1| + 3)^2 + 2(-1|(-1(-3sin(3πx - 2) + 2) + 2) - 1| + 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"
}
] | 89 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π(4(-3|(2|x + 3| - 1) + 3|) + 5) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "For the function f(x) = -1|(-2(-2sin(1π(2|x| - 1) - 1)) - 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"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = -3(-3cos(2π(3(5x^2 - 3x + 4) - 3) - 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"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 1sin(3π(2(-3|(-2|x - 2| - 1) - 2| - 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"
}
] | 39 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-4(-1|(-1|x - 3| + 1) + 3| + 3) + 1)^2 - 4(-4(-1|(-1|x - 3| + 1) + 3| + 3) + 1) + 3 negative?\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) = 2(-2cos(2π(-2|(3x - 4) + 2| + 2) + 2) + 2)^2 - 5(-2cos(2π(-2|(3x - 4) + 2| + 2) + 2) + 2) + 2 positive?\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) = 1sin(1π(3|(-3|(-5x + 1) - 1| - 3)| + 3) + 3) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "For the function f(x) = 2|(4(-1|(-1sin(3πx + 1)) - 2|)^2 - 1(-1|(-1sin(3πx + 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"
}
] | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(1|((-3(1cos(2πx - 2)) + 2)/(-2(1cos(2πx - 2)))) - 2| - 2)^2 + 5(1|((-3(1cos(2πx - 2)) + 2)/(-2(1cos(2πx - 2)))) - 2| - 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 41 | MATH |
[
{
"content": "For the function f(x) = -3sin(2π(-2|(-1(-4x) - 3) + 1| + 3) + 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"
}
] | 40 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-1cos(3π(-2(-3x^2 - 1) + 3) - 3)) + 2| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "For the function f(x) = 1(-1cos(2π(-1((2x - 3)/(2x + 2)) + 1) + 3) + 1)^2 + 1(-1cos(2π(-1((2x - 3)/(2x + 2)) + 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"
}
] | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(3π(-1((2(-4x^2 - 4x - 5) - 3)/(2(-4x^2 - 4x - 5) + 3)) - 3) + 3) is strictly decreasing.\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) = (-2(-3sin(3π(1|(-3x + 3) + 2| + 2) + 3) - 3) - 2)/(3(-3sin(3π(1|(-3x + 3) + 2| + 2) + 3) - 3) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 90 | MATH |
[
{
"content": "For the function f(x) = 1(-2cos(1π(-3(5x - 3) + 4) - 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"
}
] | 51 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-2|(3(1x^2 - 3x - 1)^2 - 1(1x^2 - 3x - 1) - 2)| + 1) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = -1|(-2(4(-4x^2 + 2)^2 - 1(-4x^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"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = 2(-1((1(1cos(1πx + 1) - 2) - 1)/(-1(1cos(1πx + 1) - 2) + 2))^2 + 4)^2 - 4(-1((1(1cos(1πx + 1) - 2) - 1)/(-1(1cos(1πx + 1) - 2) + 2))^2 + 4) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 11 | MATH |
[
{
"content": "For the function f(x) = (1(4(1(-3sin(2πx + 1) - 3) - 3)^2 - 4))/(1(4(1(-3sin(2πx + 1) - 3) - 3)^2 - 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"
}
] | 21 | MATH |
[
{
"content": "For the function f(x) = 1|(4(2cos(1π(2x^2 - 2x - 2) - 1) - 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"
}
] | 54 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(-3cos(2π(5(-1x^2 - 5x - 5)^2 + 5(-1x^2 - 5x - 5) + 1)) + 3)^2 + 2(-3cos(2π(5(-1x^2 - 5x - 5)^2 + 5(-1x^2 - 5x - 5) + 1)) + 3) + 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{You... | 8 | MATH |
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