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) = -5(-3sin(3π(2|(-3x - 4) - 2| + 1)) - 3) - 4 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) = -1(1|(1sin(1π(5x + 1)))| + 1)^2 + 3(1|(1sin(1π(5x + 1)))| + 1) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3|(-3sin(1π(1((1x - 1)/(-2x - 3)) - 2) + 1) - 3) - 2| is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 1(-2(1sin(2π(-1|x - 1| + 3) - 2) + 1) - 1)^2 + 5(-2(1sin(2π(-1|x - 1| + 3) - 2) + 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"
}
] | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-1(-2sin(3π(-2(-1x^2 + 5x + 2) + 3) + 1) + 1) + 3)/(2(-2sin(3π(-2(-1x^2 + 5x + 2) + 3) + 1) + 1) - 1) 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) = (-1(-1(-2cos(2π(3|x| - 3) - 3)) + 2))/(2(-1(-2cos(2π(3|x| - 3) - 3)) + 2) + 2) negative?\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(3cos(3π(1|((2x - 1)/(-3x - 3))| + 1) - 2) - 3) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2(2sin(2π((-3x - 1)/(3x - 2)) - 2) + 2) + 3)^2 + 3(2(2sin(2π((-3x - 1)/(3x - 2)) - 2) + 2) + 3) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = (-1(3sin(3π(1(-5x - 3) + 5) + 3) - 2) - 2)/(3(3sin(3π(1(-5x - 3) + 5) + 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"
}
] | 46 | MATH |
[
{
"content": "For the function f(x) = 3|(-2cos(2π(-2|(1x - 2) + 3| + 2) - 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"
}
] | 41 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1cos(2π(5(2|(-5x^2 + 3x + 3) - 2| + 2) - 2) - 3) + 2 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) = 2sin(2π((2(2((1x + 3)/(3x - 2)) + 1) + 1)/(-2(2((1x + 3)/(3x - 2)) + 1) + 3))) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(-3sin(2π(1|(3x^2 + 2x - 5) + 2| + 3) - 2) + 3) + 3)/(2(-3sin(2π(1|(3x^2 + 2x - 5) + 2| + 3) - 2) + 3) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "For the function f(x) = -1|(1sin(2π(1(-3x^2 - 5x - 1)^2 - 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"
}
] | 18 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1π(-2(5((-1x - 2)/(2x - 3))^2 - 5((-1x - 2)/(2x - 3)) - 5)^2 - 2(5((-1x - 2)/(2x - 3))^2 - 5((-1x - 2)/(2x - 3)) - 5) - 5) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed... | 33 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3cos(3π((1(-2|x + 2| + 3) - 1)/(-3(-2|x + 2| + 3) - 2)) + 3)) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-1sin(1π((2(-3x - 5) + 2)/(1(-3x - 5))) - 2) - 1) + 2 is strictly increasing.\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(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3)^2 - 1(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) + 4) + 3)/(1(-3(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3)^2 - 1(-1cos(1π((1x - 2)/(2x - 1)) - 3) - 3) + 4)) negative?\n\nPresent ... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(-2|(2|(2cos(2πx - 3) - 3) - 1| + 3) - 2| - 2) - 1| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(3sin(2π(3(2x - 1)^2 - 1(2x - 1) + 1)) - 2)^2 + 3(3sin(2π(3(2x - 1)^2 - 1(2x - 1) + 1)) - 2) - 3 negative?\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) = (2(3|((-3(3sin(3πx - 1) + 3) - 1)/(2(3sin(3πx - 1) + 3) - 1)) - 1| + 1))/(3(3|((-3(3sin(3πx - 1) + 3) - 1)/(2(3sin(3πx - 1) + 3) - 1)) - 1| + 1) - 2) is strictly decreasing.\n\nPresent the answer in LaTex fo... | 38 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(5((-3(1sin(3πx) + 2) + 3)/(-1(1sin(3πx) + 2) + 3)) + 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) = (1(-3cos(2π(-1(2|x + 1|) + 5) - 1) + 2) - 2)/(-3(-3cos(2π(-1(2|x + 1|) + 5) - 1) + 2)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -3(1|(2sin(3π(2|x + 1| + 2) + 1)) - 2| + 2)^2 + 1(1|(2sin(3π(2|x + 1| + 2) + 1)) - 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"
}
] | 60 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1sin(1π(-3|(5x^2 + 5x + 4) + 3| + 3) - 2) + 3) + 2| - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2sin(1π((-2(-1x - 3) + 1)/(-1(-1x - 3) + 3)) + 2) + 3)^2 - 3(-2sin(1π((-2(-1x - 3) + 1)/(-1(-1x - 3) + 3)) + 2) + 3) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 5 | MATH |
[
{
"content": "For the function f(x) = 3(-4(-2|(1sin(2πx + 1) + 1) - 3| + 2)^2 + 5(-2|(1sin(2πx + 1) + 1) - 3| + 2) + 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"
}
] | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-3(5(-2cos(2πx) - 1)^2 - 1(-2cos(2πx) - 1)) + 5) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "For the function f(x) = 2cos(2π((2((1((3x - 2)/(1x - 3)) + 1)/(1((3x - 2)/(1x - 3)))) - 3)/(1((1((3x - 2)/(1x - 3)) + 1)/(1((3x - 2)/(1x - 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}"... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(2π(-3|((3((-1x - 2)/(3x - 2)) - 1)/(-1((-1x - 2)/(3x - 2)) - 1)) + 3|) - 2) - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2cos(2π(-2|(-1(3|x - 1|)^2 + 1(3|x - 1|) - 4)| + 1) + 3) - 1 negative?\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(-2|(-1(3cos(2πx + 1) - 3)^2 - 1(3cos(2πx + 1) - 3) + 2) + 3| - 1) - 1)/(-3(-2|(-1(3cos(2πx + 1) - 3)^2 - 1(3cos(2πx + 1) - 3) + 2) + 3| - 1) + 2) is strictly increasing.\n\nPresent the answer in LaTex for... | 59 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3π(-1|(-3((2x - 1)/(3x + 1)) + 4) + 2| - 2) + 3) + 1 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) = 1|(-2|(-2sin(1π(3|x - 2|) - 3) - 1) - 3| - 2) - 2| + 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) = (2(-1cos(1π(-1|(1x - 5) - 3| - 3) + 2) + 2) - 3)/(3(-1cos(1π(-1|(1x - 5) - 3| - 3) + 2) + 2) + 1) 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) = (-1(-4(2cos(1π(5x^2 + 2x + 2) + 3) - 2)^2 + 2(2cos(1π(5x^2 + 2x + 2) + 3) - 2) - 5) - 2)/(-3(-4(2cos(1π(5x^2 + 2x + 2) + 3) - 2)^2 + 2(2cos(1π(5x^2 + 2x + 2) + 3) - 2) - 5) + 4) positive?\n\nPresent the answer in ... | 55 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1((-1(1|(1cos(1πx) + 2) - 3| + 1) + 3)/(-3(1|(1cos(1πx) + 2) - 3| + 1) + 2)) - 4 is strictly decreasing.\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|(4(-1sin(2πx + 1) - 3) - 3)| - 1)| + 3 is strictly increasing.\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(2(-4(-1cos(2πx + 1) + 1)^2 - 2)^2 - 4(-4(-1cos(2πx + 1) + 1)^2 - 2) - 1)^2 - 2(2(-4(-1cos(2πx + 1) + 1)^2 - 2)^2 - 4(-4(-1cos(2πx + 1) + 1)^2 - 2) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxe... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(-2sin(2π(-2|(1x^2 - 5) - 1| + 1) + 1) - 1) + 1 is strictly decreasing.\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) = 3sin(3π(1|((-2(-1|x| + 2))/(1(-1|x| + 2) + 3))| + 3) + 2) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = -1|(-1(-2|(3x - 1) + 1| + 3)^2 - 2(-2|(3x - 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"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = -2((-1(-3|(-3cos(1πx - 2) + 2) - 2| + 3) + 2)/(1(-3|(-3cos(1πx - 2) + 2) - 2| + 3) + 2))^2 + 1((-1(-3|(-3cos(1πx - 2) + 2) - 2| + 3) + 2)/(1(-3|(-3cos(1πx - 2) + 2) - 2| + 3) + 2)) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent ... | 39 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(-1sin(1π(-2|(-4x^2 - 3x - 5) - 1|) + 1) + 2) - 1| negative?\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|(-3|(-3|(1sin(2πx - 3) + 2) - 2| - 2)| + 3) - 1| + 3 is strictly increasing.\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((1(-1(-1sin(1πx - 3) + 2) + 3) + 2)/(-3(-1(-1sin(1πx - 3) + 2) + 3) - 2)) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -1sin(3π(-2|(3(2x + 3) + 3) + 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"
}
] | 38 | MATH |
[
{
"content": "For the function f(x) = (2((-2(1(3cos(2πx) - 1)^2 + 5(3cos(2πx) - 1) + 4) - 1)/(-1(1(3cos(2πx) - 1)^2 + 5(3cos(2πx) - 1) + 4) + 2)) - 2)/(1((-2(1(3cos(2πx) - 1)^2 + 5(3cos(2πx) - 1) + 4) - 1)/(-1(1(3cos(2πx) - 1)^2 + 5(3cos(2πx) - 1) + 4) + 2)) - 2), find the number of maximal connected intervals ... | 40 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(1|(-1(1x^2 - 1x + 1)^2 + 4(1x^2 - 1x + 1) - 2) - 1| - 2) - 3| + 1 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|(-2sin(1π(-1x + 2) + 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"
}
] | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((-1(2sin(2π(1x^2 - 2x - 3) - 2) + 3) - 3)/(2(2sin(2π(1x^2 - 2x - 3) - 2) + 3) - 1)) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3sin(1π(1|(5(1|x - 2| - 2)^2 - 3(1|x - 2| - 2) + 1) - 2|) - 3) - 3 positive?\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|(3cos(1π(-1((3x + 3)/(1x)) + 2) - 3) + 1) + 3| - 1 positive?\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) = -3|((-2(3(2cos(1πx + 2) + 1) + 4) - 1)/(-3(3(2cos(1πx + 2) + 1) + 4) - 1)) - 1| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3sin(3π(-2|(2(-2x - 5)^2 + 1) + 2| + 3) + 3) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "For the function f(x) = 4(4(-1(-3cos(3πx) - 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"
}
] | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(-1|(-2cos(1π((2x - 1)/(-1x + 3))) + 3) + 3|) + 3)/(3(-1|(-2cos(1π((2x - 1)/(-1x + 3))) + 3) + 3|) - 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) = -2|(-1sin(1π(-4((-1x + 2)/(2x + 2))^2 + 4((-1x + 2)/(2x + 2)) + 4) - 3) - 3) - 2| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "For the function f(x) = 5(-3(-3|(2sin(3πx) - 3) + 1|)^2 + 1(-3|(2sin(3πx) - 3) + 1|) - 2)^2 + 3(-3(-3|(2sin(3πx) - 3) + 1|)^2 + 1(-3|(2sin(3πx) - 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 answ... | 8 | MATH |
[
{
"content": "For the function f(x) = (-3((1(-2cos(3π(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3π(-3x - 4) - 2) - 1) + 2)) + 3)/(-2((1(-2cos(3π(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3π(-3x - 4) - 2) - 1) + 2)) + 1), find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex f... | 91 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(1|(4(3sin(3πx + 2) - 1)^2 - 5(3sin(3πx + 2) - 1)) - 1| - 2)^2 - 2(1|(4(3sin(3πx + 2) - 1)^2 - 5(3sin(3πx + 2) - 1)) - 1| - 2) + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 15 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(5(3sin(2π(-4x + 3) - 3) - 1)^2 + 1(3sin(2π(-4x + 3) - 3) - 1) + 4) - 1)/(-1(5(3sin(2π(-4x + 3) - 3) - 1)^2 + 1(3sin(2π(-4x + 3) - 3) - 1) + 4) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your... | 34 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2((-2(2sin(3π((2x - 3)/(3x + 2)) + 2) - 1) - 1)/(-2(2sin(3π((2x - 3)/(3x + 2)) + 2) - 1) - 2)) - 3)/(-1((-2(2sin(3π((2x - 3)/(3x + 2)) + 2) - 1) - 1)/(-2(2sin(3π((2x - 3)/(3x + 2)) + 2) - 1) - 2)) - 1) is s... | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2(1(-2sin(2πx)) - 4) + 4) - 3| + 2 positive?\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) = 2sin(3π(-1(-3|(-2|x + 3| - 1)| - 3) + 5)) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "For the function f(x) = 1(1sin(1π(1(-4x - 1) + 2) + 2) + 1)^2 - 3(1sin(1π(1(-4x - 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"
}
] | 69 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3cos(1π(-3(-1(2x^2 - 1x - 2)^2 + 4(2x^2 - 1x - 2) - 5) + 5) + 2) - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = (3(-3sin(3π(-2|(-2x - 2) + 3|)) - 2) + 2)/(-2(-3sin(3π(-2|(-2x - 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"
}
] | 81 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(4((1(1sin(2πx + 3) - 1))/(3(1sin(2πx + 3) - 1) - 2))^2 + 4((1(1sin(2πx + 3) - 1))/(3(1sin(2πx + 3) - 1) - 2)) + 3)| - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2π(-1(-1|(2x - 5) - 2| - 2)^2 + 2) + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2cos(3π(-2(-2(3|x + 2| + 2) + 5)^2 - 1(-2(3|x + 2| + 2) + 5) + 5) - 3) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(5(-4(-1sin(3πx - 3))^2 - 1)^2 - 1)^2 + 5(5(-4(-1sin(3πx - 3))^2 - 1)^2 - 1) + 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "For the function f(x) = 2(-3|(-1sin(1π((-3x)/(1x + 1)) - 3) + 3) + 1| - 1) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "For the function f(x) = (1(2|(-2cos(3π((-2x + 3)/(-2x - 2)) - 2) - 2) + 3| + 1) - 2)/(-3(2|(-2cos(3π((-2x + 3)/(-2x - 2)) - 2) - 2) + 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":... | 14 | MATH |
[
{
"content": "For the function f(x) = -2sin(3π(4(3(5x^2 - 4x + 5)^2 - 1(5x^2 - 4x + 5) + 4) - 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"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1|(1sin(1π(-2|((3x - 1)/(3x + 1)) + 1| + 2) - 3) + 1) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1sin(2π(-1|(3(-2|x + 3| + 3)^2 - 4(-2|x + 3| + 3) - 4) - 2| - 2)) - 3 positive?\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) = -2|(5((-3(-2cos(1πx + 3) + 3) - 3)/(3(-2cos(1πx + 3) + 3) - 2)) + 5) + 1| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = -3sin(3π(3|(-1|(3|x - 1| + 2) - 1|) + 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"
}
] | 74 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(-2|((-2(-3cos(2πx + 1) - 2) - 3)/(3(-3cos(2πx + 1) - 2)))| + 3)^2 - 4 negative?\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) = 1|(3|(2cos(1π(2x^2 - 5x + 3) + 1)) - 1|) - 3| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((-1(2|(1sin(2πx) + 1) - 3| - 2))/(-1(2|(1sin(2πx) + 1) - 3| - 2) - 1)) + 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) = (2(1((3(3sin(1πx - 2) + 1) + 2)/(2(3sin(1πx - 2) + 1) + 3))^2 + 5((3(3sin(1πx - 2) + 1) + 2)/(2(3sin(1πx - 2) + 1) + 3)) - 4) + 2)/(-2(1((3(3sin(1πx - 2) + 1) + 2)/(2(3sin(1πx - 2) + 1) + 3))^2 + 5((3(3sin(1... | 20 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(-3(3cos(3π((-1x + 3)/(2x)) - 3)) + 3) - 2)/(2(-3(3cos(3π((-1x + 3)/(2x)) - 3)) + 3)) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "For the function f(x) = 2|(-2cos(2π(-3|(-4x - 5) - 1| + 1) - 2) + 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"
}
] | 82 | MATH |
[
{
"content": "For the function f(x) = 3(-3sin(3π((2(-2x^2 - 4x - 3) + 3)/(1(-2x^2 - 4x - 3) + 3))) + 2)^2 - 4(-3sin(3π((2(-2x^2 - 4x - 3) + 3)/(1(-2x^2 - 4x - 3) + 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 answ... | 12 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(-4(-4(-2sin(3πx + 2) - 2)^2 - 1(-2sin(3πx + 2) - 2) + 4) - 2))/(1(-4(-4(-2sin(3πx + 2) - 2)^2 - 1(-2sin(3πx + 2) - 2) + 4) - 2) + 1) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed... | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3cos(3π(4(-5((-3x + 3)/(1x + 2)) + 4) + 1) - 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|((-3(1sin(1π(-3x + 5) + 1) - 2) - 2)/(-1(1sin(1π(-3x + 5) + 1) - 2) - 2))| + 1 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) = 4(1((-1(-3sin(1πx - 1) - 2) - 3)/(1(-3sin(1πx - 1) - 2) - 3)) + 1) + 5 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) = 3((-3(3sin(3π(-2|x - 3| + 2) - 1) - 2) + 3)/(1(3sin(3π(-2|x - 3| + 2) - 1) - 2) + 1))^2 - 2((-3(3sin(3π(-2|x - 3| + 2) - 1) - 2) + 3)/(1(3sin(3π(-2|x - 3| + 2) - 1) - 2) + 1)) + 4 is strictly increasing.\n\n... | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(4(1cos(3π((-2x - 3)/(-1x + 3)))) - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "For the function f(x) = 3cos(2π((1(1((2x + 2)/(-2x + 3))) + 1)/(-3(1((2x + 2)/(-2x + 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"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 1|(-3cos(3π(-1((3x - 2)/(3x + 1))^2 - 4((3x - 2)/(3x + 1)) - 4) + 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"
}
] | 18 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(-2e^(-1(3|(-2x^2 + 3x + 1)| - 2))) - 3)/(-2(-2e^(-1(3|(-2x^2 + 3x + 1)| - 2)))) 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) = -3|(4(3sin(1π(-5x^2 + 1x + 3) - 3) - 1) - 3) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "For the function f(x) = (-1(4(2sin(1π(-1|x + 3|) + 1) + 3)^2 + 5(2sin(1π(-1|x + 3|) + 1) + 3) - 1) + 2)/(2(4(2sin(1π(-1|x + 3|) + 1) + 3)^2 + 5(2sin(1π(-1|x + 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... | 11 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5(-2sin(2π(2|((-3x + 2)/(2x - 1)) - 1|) - 3) - 2)^2 + 3(-2sin(2π(2|((-3x + 2)/(2x - 1)) - 1|) - 3) - 2) - 5 positive?\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(-3(3(-1x^2 + 1x - 4) + 2) - 2) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3(2cos(3π((-3((-2x)/(2x + 3)) + 1)/(2((-2x)/(2x + 3)) - 1)) - 1) - 1) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
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