messages list | ground_truth string | dataset string |
|---|---|---|
[
{
"content": "For the function f(x) = 2cos(2π((-3(-1(1x + 3)^2 + 3(1x + 3) + 1) + 2)/(-2(-1(1x + 3)^2 + 3(1x + 3) + 1) + 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"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = 3((3(-1(3|x - 2| - 3)^2 + 3(3|x - 2| - 3) - 3))/(-2(-1(3|x - 2| - 3)^2 + 3(3|x - 2| - 3) - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -2sin(3π(4(1x + 1)^2 + 3(1x + 1) + 5) + 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"
}
] | 36 | MATH |
[
{
"content": "For the function f(x) = 2sin(2π(2((-3(1x^2 - 2x - 2) + 2)/(2(1x^2 - 2x - 2) + 2)) - 5) - 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"
}
] | 15 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1π(5(2x - 1)^2 + 3(2x - 1)) - 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(3((-1x + 2)/(2x + 3)) - 2)^2 - 1(3((-1x + 2)/(2x + 3)) - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = (-1(2cos(1π(2(1x^2 - 4) + 1) - 2) - 3))/(-3(2cos(1π(2(1x^2 - 4) + 1) - 2) - 3) + 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "For the function f(x) = -2|(3((-3x - 1)/(1x + 1))^2 + 1((-3x - 1)/(1x + 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"
}
] | 3 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3(3(3cos(2π((1x + 2)/(-3x + 4)) - 3) + 2) - 4)^2 - 2(3(3cos(2π((1x + 2)/(-3x + 4)) - 3) + 2) - 4) - 1)| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "For the function f(x) = (3(1(-1|(2x^2 + 5x - 4) + 1| + 2)) + 2)/(-3(1(-1|(2x^2 + 5x - 4) + 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = (-3(-3(3cos(3π(3|x + 3| + 3) - 2) + 2) - 3) + 1)/(-2(-3(3cos(3π(3|x + 3| + 3) - 2) + 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"
}
] | 78 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2e^(-1((-3(1(-1|(1x - 3) - 3|)^2 + 2(-1|(1x - 3) - 3|) - 5) + 2)/(2(1(-1|(1x - 3) - 3|)^2 + 2(-1|(1x - 3) - 3|) - 5) - 1))) + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(1|(1(3sin(1π((2x - 2)/(2x - 3)) - 3)) - 2) + 2| + 2)^2 + 4(1|(1(3sin(1π((2x - 2)/(2x - 3)) - 3)) - 2) + 2| + 2) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 4 | MATH |
[
{
"content": "For the function f(x) = -1(-2|((-1(1sin(1πx - 1) + 2) - 1)/(-3(1sin(1πx - 1) + 2))) - 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"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = -1(-5(-2|x - 3| + 1) + 1)^2 - 3(-5(-2|x - 3| + 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"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(4(-3|x - 2| + 3)^2 - 2) - 3 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) = (-3(2|(-3sin(1π(1(3x - 3)^2 + 2) + 2) - 3) + 3| + 2) + 3)/(-2(2|(-3sin(1π(1(3x - 3)^2 + 2) + 2) - 3) + 3| + 2) + 3) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 44 | MATH |
[
{
"content": "For the function f(x) = -1cos(1π(1|((1x + 1)/(1x)) - 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"
}
] | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(5(2|(-3cos(1πx - 1) + 2)| - 2) + 2)^2 - 5(5(2|(-3cos(1πx - 1) + 2)| - 2) + 2) - 4 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((-2(-1(-2|x + 1| + 3) + 1) + 1)/(1(-1(-2|x + 1| + 3) + 1) + 3))^2 + 3((-2(-1(-2|x + 1| + 3) + 1) + 1)/(1(-1(-2|x + 1| + 3) + 1) + 3)) + 2 positive?\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) = -5(3|((-3(-4x^2 + 4x + 2))/(1(-4x^2 + 4x + 2) - 3))| - 3) - 4 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((1(1x - 4) - 3)/(-2(1x - 4) - 2))^2 + 5((1(1x - 4) - 3)/(-2(1x - 4) - 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"
}
] | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(1((-3x - 2)/(3x - 2))^2 - 2((-3x - 2)/(3x - 2)) + 2) - 2| + 1 is strictly increasing.\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) = -2((3(3(2e^(2(3|x + 1| - 3)))^2 - 3(2e^(2(3|x + 1| - 3))) + 5) - 1)/(-1(3(2e^(2(3|x + 1| - 3)))^2 - 3(2e^(2(3|x + 1| - 3))) + 5) + 2)) - 1 is strictly increasing.\n\nPresent the answer in LaTex format: \\box... | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(1cos(1π(2((-2x + 2)/(-2x + 3))^2 + 3((-2x + 2)/(-2x + 3)) - 5) + 1) + 3) - 3 is strictly decreasing.\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|(-3((2x + 1)/(-1x + 2))^2 + 4((2x + 1)/(-1x + 2)) + 5) + 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) = -3(1cos(2π((2(1|(2x^2 - 4x) + 1| + 2) + 1)/(-1(1|(2x^2 - 4x) + 1| + 2) - 3)) - 1) - 2) + 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) = (-1(1sin(1π(-5(3|x - 2| + 1)^2 + 2(3|x - 2| + 1) + 5) - 3) - 3) - 2)/(3(1sin(1π(-5(3|x - 2| + 1)^2 + 2(3|x - 2| + 1) + 5) - 3) - 3) - 1) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 41 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(4((1x + 2)/(3x - 2)) - 5)^2 + 2(4((1x + 2)/(3x - 2)) - 5) + 4 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(-3cos(3π((2x + 3)/(-1x - 3)) + 3) + 1) - 5) - 1| + 1 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) = 2|(1((-2x + 2)/(1x))^2 - 5((-2x + 2)/(1x)) + 4) - 3| + 3 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -1e^(1(-4(1x^2 + 2x + 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) = 1|(-1(-1cos(1πx - 1) - 1) + 5) - 3| + 1 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) = 2((2(-1|(-3x) - 1| + 3) - 2)/(-1(-1|(-3x) - 1| + 3) + 2))^2 - 5((2(-1|(-3x) - 1| + 3) - 2)/(-1(-1|(-3x) - 1| + 3) + 2)) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 3 | MATH |
[
{
"content": "For the function f(x) = 2cos(1π(-1((-1(1|x - 1| + 3) + 2)/(3(1|x - 1| + 3) - 1))^2 - 4((-1(1|x - 1| + 3) + 2)/(3(1|x - 1| + 3) - 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": "us... | 2 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(2|(1(1x^2 + 4x - 3) - 3) - 1| + 3) + 3)/(2(2|(1(1x^2 + 4x - 3) - 3) - 1| + 3) - 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) = (1(3sin(3π(1|x + 1| - 1) - 2) + 2) - 3)/(2(3sin(3π(1|x + 1| - 1) - 2) + 2) + 3) 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) = 3((1(-1|(-3x - 1) - 3| + 3) - 3)/(2(-1|(-3x - 1) - 3| + 3) + 1))^2 + 2((1(-1|(-3x - 1) - 3| + 3) - 3)/(2(-1|(-3x - 1) - 3| + 3) + 1)) + 3 positive?\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) = 3|(-3((-2x)/(-3x + 2))^2 - 1) + 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) = 2((3(-2|(4x^2 - 4x - 5) - 3| + 3) - 3)/(2(-2|(4x^2 - 4x - 5) - 3| + 3) + 3)) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1sin(1π(-5(2|((-2x - 2)/(-2x - 3)) + 3| - 2) - 4) - 1) + 2 positive?\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) = -2sin(2π(2|(-1x - 4) + 2| - 1) - 2) + 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) = 1e^(-2(1((1(-2|(2x^2 - 2x - 4) - 1| + 3) - 3)/(3(-2|(2x^2 - 2x - 4) - 1| + 3) + 1)) - 4)) + 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) = -1(-1(-1cos(1π((2(1|x + 3| - 1) - 1)/(-3(1|x + 3| - 1) - 2)) - 3) - 1) + 4)^2 - 3(-1(-1cos(1π((2(1|x + 3| - 1) - 1)/(-3(1|x + 3| - 1) - 2)) - 3) - 1) + 4) + 1 is strictly increasing.\n\nPresent the answer in... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-2|(3((3(-1sin(1πx) - 2) + 1)/(-2(-1sin(1πx) - 2) - 2)) + 5) - 3| + 2)^2 + 5 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) = -1cos(3π(-5(2((-2x - 2)/(1x + 2)) - 1)^2 - 1(2((-2x - 2)/(1x + 2)) - 1) - 4) + 2) + 3 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) = (2(3|(5(3e^(1(-4x^2 + 5x - 2)) + 1) + 5) - 2| - 1))/(-2(3|(5(3e^(1(-4x^2 + 5x - 2)) + 1) + 5) - 2| - 1) - 1) 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) = -3((-2(2|(3(2sin(2πx + 3) + 3) + 2) + 3| + 3) - 2)/(1(2|(3(2sin(2πx + 3) + 3) + 2) + 3| + 3)))^2 - 2((-2(2|(3(2sin(2πx + 3) + 3) + 2) + 3| + 3) - 2)/(1(2|(3(2sin(2πx + 3) + 3) + 2) + 3| + 3))) - 3 is strictl... | 21 | MATH |
[
{
"content": "For the function f(x) = 5(1|(1x - 3) - 2| - 2)^2 + 4(1|(1x - 3) - 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"
}
] | 2 | MATH |
[
{
"content": "For the function f(x) = -1e^(-1(2(-3|(4((2x - 3)/(2x)) - 2) - 1|)^2 + 5(-3|(4((2x - 3)/(2x)) - 2) - 1|) + 1)) - 1, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
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