messages listlengths 1 1 | ground_truth stringclasses 70
values | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2π(-4x^2 - 4) + 2) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(2|(-2cos(1πx + 2) - 1) + 3| + 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = 2(2((2(-3|x - 3|) + 3)/(1(-3|x - 3|) - 2))^2 - 5((2(-3|x - 3|) + 3)/(1(-3|x - 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"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(2cos(1πx + 1) - 1) + 2)/(2(2cos(1πx + 1) - 1) + 2) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = (-2(2cos(2πx + 2) - 3))/(-3(2cos(2πx + 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(3x^2 + 3x - 3) - 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"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-1(1|(-1sin(3πx - 3) + 2) - 1| + 3) + 3)/(-2(1|(-1sin(3πx - 3) + 2) - 1| + 3) - 1) is strictly decreasing.\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) = -3((2(-3(-3|x + 3|) - 5) + 3)/(-1(-3(-3|x + 3|) - 5)))^2 - 2((2(-3(-3|x + 3|) - 5) + 3)/(-1(-3(-3|x + 3|) - 5))) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 3 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(-2|(-5x + 2) + 3| + 3)^2 - 3(-2|(-5x + 2) + 3| + 3) + 3 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) = 1cos(1π((-1x + 3)/(1x)) + 3) + 3 positive?\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(4(3|(5x^2 + 4x - 5) + 3| + 3) + 1) + 3)/(3(4(3|(5x^2 + 4x - 5) + 3| + 3) + 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) = 5(3|(-3((2x + 3)/(-3x + 2)) - 2) - 1| - 1)^2 + 1(3|(-3((2x + 3)/(-3x + 2)) - 2) - 1| - 1) - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = 1|(1cos(1πx - 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"
}
] | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1|(-5(5((-2x - 2)/(1x + 2)))^2 + 4(5((-2x - 2)/(1x + 2))) + 3) - 1| + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -2|(-2((2x + 2)/(2x - 1))^2 + 2((2x + 2)/(2x - 1)) - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = (3(-2|(2(-1cos(3πx + 1) + 3)^2 + 1(-1cos(3πx + 1) + 3))| + 1) + 1)/(3(-2|(2(-1cos(3πx + 1) + 3)^2 + 1(-1cos(3πx + 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}",
... | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 3e^(1(3((-1x)/(2x - 1))^2 + 2)) + 1 is strictly decreasing.\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(4((1(2|x - 3| + 1) + 3)/(3(2|x - 3| + 1) - 2)) - 3)^2 - 4(4((1(2|x - 3| + 1) + 3)/(3(2|x - 3| + 1) - 2)) - 3) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 4(-1|(2sin(2πx - 1) + 2) - 3| + 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": "For the function f(x) = -5(1cos(1πx + 2) + 2)^2 - 5(1cos(1πx + 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"
}
] | 11 | MATH |
[
{
"content": "For the function f(x) = -2cos(1π(2(-1|((-3x - 3)/(3x - 1)) + 1| + 2)^2 - 3(-1|((-3x - 3)/(3x - 1)) + 1| + 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"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = 4(-1(3|((3x)/(-1x - 2)) + 2| - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = (-1(-1cos(1π(-2(3|x - 3| - 2)^2 + 5(3|x - 3| - 2) + 2) + 3)) - 1)/(1(-1cos(1π(-2(3|x - 3| - 2)^2 + 5(3|x - 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}",
"ro... | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((2(-2|x - 3| + 2) - 1)/(-3(-2|x - 3| + 2) - 2))^2 + 4((2(-2|x - 3| + 2) - 1)/(-3(-2|x - 3| + 2) - 2)) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(-1(-3|x - 2| + 1)^2 + 1(-3|x - 2| + 1) - 1) + 2 negative?\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((2(2cos(3π(-4x + 3) + 2) - 2) + 1)/(-1(2cos(3π(-4x + 3) + 2) - 2) + 3))^2 + 2((2(2cos(3π(-4x + 3) + 2) - 2) + 1)/(-1(2cos(3π(-4x + 3) + 2) - 2) + 3)) + 4 is strictly increasing.\n\nPresent the answer in La... | 42 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π(-5(-1|(-1x^2 - 1x - 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"
}
] | 45 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(2|(2cos(2πx - 2) - 2) - 1| - 2)^2 - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3|(-3e^(-1(-2((-1x - 2)/(3x))^2 - 1((-1x - 2)/(3x)) - 5)) - 2) - 1| + 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 3cos(3π(-3|((1x + 3)/(1x)) - 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2sin(2π(-2|x - 2| + 2)) + 2 is strictly decreasing.\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(4((-2(-2|x + 2| + 3) + 2)/(-2(-2|x + 2| + 3) + 1))^2 - 4) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(2cos(2πx + 3) + 1) + 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) = -3cos(1π(1(5x^2 + 2x - 5) - 4) - 2) is strictly increasing.\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(3πx + 3) - 3))/(2(2cos(3πx + 3) - 3) - 1) negative?\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) = -2|(1sin(2π((1x)/(3x - 2)) + 3)) - 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) = 2|((-2x)/(1x + 2)) - 1| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -1|((-3(1x + 4) + 2)/(1(1x + 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(3sin(2π(2x - 4) - 2) + 2) - 1)/(3(3sin(2π(2x - 4) - 2) + 2) - 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) = 2cos(2π(-5x + 2) + 1) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "For the function f(x) = 2|((-2(1x + 1) + 3)/(-3(1x + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(5((1x + 1)/(3x - 2)) - 1)| + 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 1((-3(1sin(3πx + 1) + 2) + 1)/(-3(1sin(3πx + 1) + 2) + 2))^2 + 5((-3(1sin(3πx + 1) + 2) + 1)/(-3(1sin(3πx + 1) + 2) + 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}",
"rol... | 31 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|((-2(1x^2 + 3x - 2))/(-1(1x^2 + 3x - 2) - 1)) + 2| - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1sin(3π(1((-3(4x^2 + 1x - 2))/(2(4x^2 + 1x - 2) + 1)) + 1) + 2) - 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "For the function f(x) = 1sin(2π(4x - 4) - 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"
}
] | 81 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2cos(3π((2x - 2)/(-1x - 2)) + 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) = -4((1(-3(-3|x - 3| + 2) + 3) - 1)/(-2(-3(-3|x - 3| + 2) + 3) + 1))^2 - 4((1(-3(-3|x - 3| + 2) + 3) - 1)/(-2(-3(-3|x - 3| + 2) + 3) + 1)) + 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 2 | MATH |
[
{
"content": "For the function f(x) = 3|(-1e^(-2(-2(-3x^2 + 1x) - 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"
}
] | 3 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(-2((2(1x) + 1)/(1(1x) - 2))^2 - 1((2(1x) + 1)/(1(1x) - 2)) + 2) + 3| - 3 is strictly decreasing.\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(3(-2sin(2π(-1|x| - 2) - 3) + 1)^2 - 2(-2sin(2π(-1|x| - 2) - 3) + 1) + 3) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -5(1|(1x^2 + 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((-1(5(-1x - 1)^2 + 2(-1x - 1) - 4) - 2)/(-3(5(-1x - 1)^2 + 2(-1x - 1) - 4) + 2)) + 1| positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(3cos(1π(-1((-2x - 1)/(-3x + 1)) + 5) + 2) - 2)^2 + 4(3cos(1π(-1((-2x - 1)/(-3x + 1)) + 5) + 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "For the function f(x) = (2(-1|(-2sin(2πx - 1) + 3)| - 2))/(-1(-1|(-2sin(2πx - 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"
}
] | 20 | MATH |
[
{
"content": "For the function f(x) = -2|(3(-1x^2 - 5x + 3) - 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -5((1(5(1cos(2πx - 2) + 2)) + 1)/(-3(5(1cos(2πx - 2) + 2)) - 1))^2 + 1((1(5(1cos(2πx - 2) + 2)) + 1)/(-3(5(1cos(2πx - 2) + 2)) - 1)) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3cos(1π(-1|x - 1| - 2) + 1) - 3 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) = (-2(-1sin(3πx + 2) - 1) - 2)/(-3(-1sin(3πx + 2) - 1) - 2) positive?\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) = -3|(1x^2 - 2x - 1) - 1| - 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) = 2cos(3π((-1(1x^2 + 2x - 3) + 3)/(-1(1x^2 + 2x - 3))) + 2) + 3 is strictly increasing.\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) = 1|((2(-2(1x^2 - 5x - 3) - 3) + 1)/(-2(-2(1x^2 - 5x - 3) - 3) + 3)) + 3| - 3 is strictly decreasing.\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(-1|((-3(2cos(3πx + 2) - 1) + 3)/(-3(2cos(3πx + 2) - 1) - 3)) + 2|)^2 + 1(-1|((-3(2cos(3πx + 2) - 1) + 3)/(-3(2cos(3πx + 2) - 1) - 3)) + 2|) - 3 is strictly increasing.\n\nPresent the answer in LaTex forma... | 42 | MATH |
[
{
"content": "For the function f(x) = (1(-2sin(3π(3(-3x + 5)^2 - 3(-3x + 5)) + 3) - 3) + 2)/(-2(-2sin(3π(3(-3x + 5)^2 - 3(-3x + 5)) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1sin(1π(-3((1x - 1)/(-2x + 1)) - 1) - 2) - 2)^2 + 2(1sin(1π(-3((1x - 1)/(-2x + 1)) - 1) - 2) - 2) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = 2|(-1sin(3π((3(1x^2 - 5x + 2))/(3(1x^2 - 5x + 2) + 2)) - 1) + 1) - 1| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(4((-2(-2sin(1πx + 1)) - 2)/(-1(-2sin(1πx + 1)))) + 3)^2 + 3(4((-2(-2sin(1πx + 1)) - 2)/(-1(-2sin(1πx + 1)))) + 3) - 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 23 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(3π(-2|(4x + 1)| - 3) + 1) + 3 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|((1(-3(2x - 4)^2 - 4(2x - 4) + 2) + 3)/(2(-3(2x - 4)^2 - 4(2x - 4) + 2) + 3)) - 1| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = -1cos(1π(2(-1x^2 - 5) + 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"
}
] | 51 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2|((-2(-1(2x^2 - 5x + 1) + 5) + 2)/(-1(-1(2x^2 - 5x + 1) + 5) - 3)) + 3| + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(2sin(1π(3|x - 1| + 1) + 3) + 1)^2 - 3(2sin(1π(3|x - 1| + 1) + 3) + 1) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1e^(2(-4(5x^2 + 1x - 5) - 3)) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -1(-3(2sin(3πx + 2) - 1) - 4)^2 - 2(-3(2sin(3πx + 2) - 1) - 4) - 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"
}
] | 61 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(5((3x - 3)/(-3x + 2)) + 3) - 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) = 1|(-3sin(3π((-3x - 1)/(2x + 2)) - 1) - 2) - 2| + 3 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(4(-1|x - 1|) + 5)^2 - 5(4(-1|x - 1|) + 5) - 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 4(-2|(-2cos(3π((1x - 2)/(-1x + 3)) + 2) + 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"
}
] | 7 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(1|(3((-2x - 3)/(3x - 2)) + 5) + 1| - 3)^2 - 1(1|(3((-2x - 3)/(3x - 2)) + 5) + 1| - 3) + 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4((3(2|x| - 2) + 3)/(1(2|x| - 2)))^2 + 5((3(2|x| - 2) + 3)/(1(2|x| - 2))) + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(1|((-1(3e^(2x) - 3) - 2)/(-1(3e^(2x) - 3) + 2)) + 3| - 2)^2 + 5(1|((-1(3e^(2x) - 3) - 2)/(-1(3e^(2x) - 3) + 2)) + 3| - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 2 | MATH |
[
{
"content": "For the function f(x) = -1sin(2π((2x - 3)/(-2x - 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"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = -3(1cos(2π((-1x - 2)/(-1x - 3)) - 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"
}
] | 6 | MATH |
[
{
"content": "For the function f(x) = 1cos(2π(-2|x + 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"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = -3sin(1π(-4(2|x - 2| + 3)^2 - 5(2|x - 2| + 3) - 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"
}
] | 22 | MATH |
[
{
"content": "For the function f(x) = -4(3|((-2(2sin(2πx + 3) + 3) - 1)/(1(2sin(2πx + 3) + 3))) - 3| - 3) - 5, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1((3(1|x| - 2) - 3)/(2(1|x| - 2) + 1))^2 - 3((3(1|x| - 2) - 3)/(2(1|x| - 2) + 1)) - 5 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -4(3cos(3π((-2x - 1)/(-3x - 2)) + 1) - 2)^2 - 1(3cos(3π((-2x - 1)/(-3x - 2)) + 1) - 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) = 3sin(1π(-5((-3(4x - 1) - 2)/(-3(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"
}
] | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(-2(4(3cos(3πx - 2) - 1) - 4)^2 + 4(4(3cos(3πx - 2) - 1) - 4) + 4) + 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((-1x + 3)/(-1x + 2))^2 + 5((-1x + 3)/(-1x + 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) = 2|(-3sin(1πx - 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"
}
] | 10 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3((2(-3|(-3x^2 - 4x + 3) - 2| - 1) - 3)/(2(-3|(-3x^2 - 4x + 3) - 2| - 1) + 1)) + 4 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) = -1e^(-2(2(-4x)^2 + 4(-4x) + 2)) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 1(-1e^(-1((-2x - 3)/(-2x - 1))) - 1)^2 + 3(-1e^(-1((-2x - 3)/(-2x - 1))) - 1) - 4, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = 2|(-1((-2x + 2)/(1x + 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -3sin(2π((3(1(2|x - 1| - 3)^2 - 1(2|x - 1| - 3) + 4))/(3(1(2|x - 1| - 3)^2 - 1(2|x - 1| - 3) + 4) + 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": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -3sin(2π(-4(-3x - 4)^2 + 1(-3x - 4) - 4) + 3) + 1 is strictly increasing.\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|(4(4((1x)/(-2x + 1)) + 1)^2 + 2) + 2| - 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) = 4(2cos(1πx + 3) - 2)^2 - 2(2cos(1πx + 3) - 2) + 4 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
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