messages listlengths 1 1 | ground_truth stringclasses 76 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2((1(-3sin(1π((-3(-4x^2 + 5x + 3) + 1)/(-2(-4x^2 + 5x + 3) - 3)) - 2) + 1) - 3)/(1(-3sin(1π((-3(-4x^2 + 5x + 3) + 1)/(-2(-4x^2 + 5x + 3) - 3)) - 2) + 1) - 2)) + 2)/(2((1(-3sin(1π((-3(-4x^2 + 5x + 3) + 1)/(-2(-4x^... | 5 | MATH |
[
{
"content": "For the function f(x) = -2((1(5(5(-2cos(2πx - 1) - 2)^2 - 1(-2cos(2πx - 1) - 2) + 5) - 4) + 2)/(-2(5(5(-2cos(2πx - 1) - 2)^2 - 1(-2cos(2πx - 1) - 2) + 5) - 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... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(4(1|(-2((3x + 3)/(3x - 2))^2 - 5((3x + 3)/(3x - 2))) + 3| - 2)^2 - 2)| 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) = 4(-4(2cos(1π(5(2|x + 1|)^2 + 5(2|x + 1|)) - 2) - 2)^2 - 5) + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = 2sin(3π(3(2|(1|((-3x + 3)/(-1x - 3)) + 2| - 1)| - 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"
}
] | 43 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -1(2cos(2π(-1((2(4x) + 3)/(-3(4x) + 2))^2 + 3((2(4x) + 3)/(-3(4x) + 2)) - 1) - 1)) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "For the function f(x) = (1(-1(-2sin(2π(2|(-2x^2 - 3x + 2) + 3| + 1) + 2) - 1) + 4) + 2)/(2(-1(-2sin(2π(2|(-2x^2 - 3x + 2) + 3| + 1) + 2) - 1) + 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}",
"r... | 53 | MATH |
[
{
"content": "For the function f(x) = (-2(-1(4(-2((3x - 3)/(-1x))^2 + 2((3x - 3)/(-1x)) + 4)^2 + 5(-2((3x - 3)/(-1x))^2 + 2((3x - 3)/(-1x)) + 4) + 1)^2 + 2(4(-2((3x - 3)/(-1x))^2 + 2((3x - 3)/(-1x)) + 4)^2 + 5(-2((3x - 3)/(-1x))^2 + 2((3x - 3)/(-1x)) + 4) + 1)))/(-2(-1(4(-2((3x - 3)/(-1x))^2 + 2((3x - 3)/(-1x))... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-1|(-4((-1(1cos(2πx + 2)) - 1)/(-1(1cos(2πx + 2)) - 2)) - 2) + 3| + 2) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "For the function f(x) = 1|(1(4(3|(-3sin(3πx - 2) - 1) - 3| + 2)^2 + 5(3|(-3sin(3πx - 2) - 1) - 3| + 2) - 2) - 4) + 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": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(2π((-2((-1(-1(-2|x| + 2) + 1) + 1)/(1(-1(-2|x| + 2) + 1))) + 1)/(-3((-1(-1(-2|x| + 2) + 1) + 1)/(1(-1(-2|x| + 2) + 1))))) + 1) - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 5 | MATH |
[
{
"content": "For the function f(x) = (2(1(-3(-1(2sin(1πx + 3))^2 + 5(2sin(1πx + 3)) + 3)^2 - 4(-1(2sin(1πx + 3))^2 + 5(2sin(1πx + 3)) + 3) + 2)^2 + 5(-3(-1(2sin(1πx + 3))^2 + 5(2sin(1πx + 3)) + 3)^2 - 4(-1(2sin(1πx + 3))^2 + 5(2sin(1πx + 3)) + 3) + 2) + 1) - 1)/(1(1(-3(-1(2sin(1πx + 3))^2 + 5(2sin(1πx + 3)) + ... | 19 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2|((1((2(2sin(2πx - 3) + 3))/(-1(2sin(2πx - 3) + 3) - 3)) + 3)/(2((2(2sin(2πx - 3) + 3))/(-1(2sin(2πx - 3) + 3) - 3)) + 3)) + 3| + 1)| + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (1(5(2(3(-3sin(2πx - 3) - 1) - 4))^2 + 5(2(3(-3sin(2πx - 3) - 1) - 4)) - 1))/(3(5(2(3(-3sin(2πx - 3) - 1) - 4))^2 + 5(2(3(-3sin(2πx - 3) - 1) - 4)) - 1) + 2) negative?\n\nPresent the answer in LaTex format: \\boxe... | 41 | MATH |
[
{
"content": "For the function f(x) = -2cos(1π(-5(-4(1|((2x - 3)/(2x + 3)) - 2|)^2 - 3(1|((2x - 3)/(2x + 3)) - 2|) - 5) - 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"
}
] | 39 | MATH |
[
{
"content": "For the function f(x) = 1cos(1π(1|(2(-4(3x - 4) + 2)^2 + 3) + 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"
}
] | 10 | MATH |
[
{
"content": "For the function f(x) = 1((1(-1(1sin(1π((2x - 3)/(-3x - 1)) - 3) - 2)^2 + 4(1sin(1π((2x - 3)/(-3x - 1)) - 3) - 2) + 5) + 1)/(2(-1(1sin(1π((2x - 3)/(-3x - 1)) - 3) - 2)^2 + 4(1sin(1π((2x - 3)/(-3x - 1)) - 3) - 2) + 5) - 1))^2 + 4((1(-1(1sin(1π((2x - 3)/(-3x - 1)) - 3) - 2)^2 + 4(1sin(1π((2x - 3)/(-... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-4(1(-1|(2|x + 2|) - 3|) + 1)^2 + 1) 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) = -1(1|(-2(-3(2x^2 + 1x - 3)^2 + 1(2x^2 + 1x - 3) + 3) + 3) - 3| + 2)^2 + 1(1|(-2(-3(2x^2 + 1x - 3)^2 + 1(2x^2 + 1x - 3) + 3) + 3) - 3| + 2) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(4(1sin(3π((3(5x^2 - 4x - 3) + 2)/(2(5x^2 - 4x - 3) + 3))) + 2) + 5) + 1)/(-1(4(1sin(3π((3(5x^2 - 4x - 3) + 2)/(2(5x^2 - 4x - 3) + 3))) + 2) + 5) + 3) negative?\n\nPresent the answer in LaTex format: \\boxed{You... | 5 | MATH |
[
{
"content": "For the function f(x) = -3((2((-2(1|(-1|x - 1|) + 2| - 3) + 3)/(3(1|(-1|x - 1|) + 2| - 3) - 1)) - 3)/(1((-2(1|(-1|x - 1|) + 2| - 3) + 3)/(3(1|(-1|x - 1|) + 2| - 3) - 1)) + 2))^2 - 5((2((-2(1|(-1|x - 1|) + 2| - 3) + 3)/(3(1|(-1|x - 1|) + 2| - 3) - 1)) - 3)/(1((-2(1|(-1|x - 1|) + 2| - 3) + 3)/(3(1|(... | 4 | MATH |
[
{
"content": "For the function f(x) = 5(-4(-1|((-3(3cos(2πx + 2) + 1) + 3)/(2(3cos(2πx + 2) + 1) + 2)) + 3| - 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"
}
] | 30 | MATH |
[
{
"content": "For the function f(x) = (1((3(4(-1|(3cos(2πx + 1) + 1) - 3| - 2) - 3) - 1)/(-3(4(-1|(3cos(2πx + 1) + 1) - 3| - 2) - 3) + 2)) + 1)/(-3((3(4(-1|(3cos(2πx + 1) + 1) - 3| - 2) - 3) - 1)/(-3(4(-1|(3cos(2πx + 1) + 1) - 3| - 2) - 3) + 2)) + 2), find the number of maximal connected intervals in [-10, 10] ... | 41 | MATH |
[
{
"content": "For the function f(x) = 3(-1cos(1π(-4(-1|(-2x + 4) - 2| + 2) + 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"
}
] | 81 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(-2|(-3cos(2π(-3|(2x - 2) + 2| + 3) - 2) + 2) - 3| + 2) - 1)/(2(-2|(-3cos(2π(-3|(2x - 2) + 2| + 3) - 2) + 2) - 3| + 2) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 44 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2|(3((-3(-2|(1x^2 + 4x - 4) + 1| + 2))/(1(-2|(1x^2 + 4x - 4) + 1| + 2) - 2))^2 + 5((-3(-2|(1x^2 + 4x - 4) + 1| + 2))/(1(-2|(1x^2 + 4x - 4) + 1| + 2) - 2)) + 4) - 3| - 2 is strictly decreasing.\n\nPresent th... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1|(-1|(2sin(1π(-3x) - 1) + 1) - 3| - 3) + 2| + 2) + 1| - 2 is strictly decreasing.\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) = -2|(2sin(3π(2|((-2(2x^2 + 1))/(2(2x^2 + 1) - 1)) - 3| + 3) - 3) + 2) - 1| - 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2|(1sin(1π((-3(2x^2 - 3x - 1) - 3)/(-1(2x^2 - 3x - 1) + 2)) - 1) - 1) + 2| + 3)^2 + 2(2|(1sin(1π((-3(2x^2 - 3x - 1) - 3)/(-1(2x^2 - 3x - 1) + 2)) - 1) - 1) + 2| + 3) - 1 positive?\n\nPresent the answer in LaTex ... | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((2(3|(2sin(2π(2x^2 + 3x + 5) - 1) + 3)| + 2) + 2)/(2(3|(2sin(2π(2x^2 + 3x + 5) - 1) + 3)| + 2) - 1)) - 3| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|((1(-3sin(2π(1(1|x - 1| - 1) + 1))) - 2)/(2(-3sin(2π(1(1|x - 1| - 1) + 1))) - 1)) - 2| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1(-5((3((1(-3sin(1πx - 3) - 1) - 3)/(1(-3sin(1πx - 3) - 1) + 1)) - 1)/(3((1(-3sin(1πx - 3) - 1) - 3)/(1(-3sin(1πx - 3) - 1) + 1))))^2 - 1((3((1(-3sin(1πx - 3) - 1) - 3)/(1(-3sin(1πx - 3) - 1) + 1)) - 1)/(3((... | 10 | MATH |
[
{
"content": "For the function f(x) = -1cos(2π(5((1(4(1x)^2 + 1(1x)))/(3(4(1x)^2 + 1(1x)) + 3))^2 + 4((1(4(1x)^2 + 1(1x)))/(3(4(1x)^2 + 1(1x)) + 3)) - 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... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2cos(2π(-3|((-1((-2(-3|x - 3| + 2) - 1)/(-1(-3|x - 3| + 2) - 1)) + 1)/(1((-2(-3|x - 3| + 2) - 1)/(-1(-3|x - 3| + 2) - 1))))| + 2) - 1) - 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(1sin(2π(-3(-3(2|x + 2| + 1) + 3))) - 2))/(2(1sin(2π(-3(-3(2|x + 2| + 1) + 3))) - 2) - 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-2(3|(-2|(1cos(3π(-1|x - 2| + 3)) + 1) + 1| - 1)| + 1) + 3)/(2(3|(-2|(1cos(3π(-1|x - 2| + 3)) + 1) + 1| - 1)| + 1) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 30 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(1((1(-2sin(1π((3x - 1)/(-3x)) + 3)) + 1)/(2(-2sin(1π((3x - 1)/(-3x)) + 3))))^2 + 5((1(-2sin(1π((3x - 1)/(-3x)) + 3)) + 1)/(2(-2sin(1π((3x - 1)/(-3x)) + 3)))) + 2) + 2 negative?\n\nPresent the answer in LaTex for... | 6 | MATH |
[
{
"content": "For the function f(x) = -1|((3(5(-4(2cos(2πx + 1) - 3)^2 + 2(2cos(2πx + 1) - 3) + 5)) - 3)/(3(5(-4(2cos(2πx + 1) - 3)^2 + 2(2cos(2πx + 1) - 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}"... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1π((2(1|(2|(-1x^2 - 2x - 1) + 3| - 2) - 1| + 1) + 1)/(-1(1|(2|(-1x^2 - 2x - 1) + 3| - 2) - 1| + 1) + 3)) + 3) + 3 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 15 | MATH |
[
{
"content": "For the function f(x) = 1(3sin(2π(5(1(-2x^2 - 5x - 5)^2 + 3(-2x^2 - 5x - 5) - 1) + 3)) + 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"
}
] | 9 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2|(3cos(1π(-1(4x - 3) - 1) + 1) + 2) + 3| - 2)^2 - 5(2|(3cos(1π(-1(4x - 3) - 1) + 1) + 2) + 3| - 2) - 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(-1cos(1π(1(-5x^2 + 3x + 5) - 2)) + 2) + 1| + 3) + 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "For the function f(x) = -1(-2sin(2π(2(-1|(2x^2 - 1x - 1) + 3| + 1)^2 - 5(-1|(2x^2 - 1x - 1) + 3| + 1)) + 1) + 3)^2 + 4(-2sin(2π(2(-1|(2x^2 - 1x - 1) + 3| + 1)^2 - 5(-1|(2x^2 - 1x - 1) + 3| + 1)) + 1) + 3) - 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent ... | 9 | MATH |
[
{
"content": "For the function f(x) = (2(1cos(1π(1((-2(-4x - 5) + 3)/(1(-4x - 5))) - 4) + 1) - 2) + 3)/(-2(1cos(1π(1((-2(-4x - 5) + 3)/(1(-4x - 5))) - 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}",
... | 4 | MATH |
[
{
"content": "For the function f(x) = -2(2|(-1|(-4(2sin(3πx) - 3) - 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"
}
] | 31 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(2(-1(4((1x - 2)/(-3x - 1))^2 - 2((1x - 2)/(-3x - 1)) - 5) - 5)^2 + 3(-1(4((1x - 2)/(-3x - 1))^2 - 2((1x - 2)/(-3x - 1)) - 5) - 5))^2 - 2(2(-1(4((1x - 2)/(-3x - 1))^2 - 2((1x - 2)/(-3x - 1)) - 5) - 5)^2 + 3(-1(4(... | 5 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (3(1cos(1π(-1(-2|(-1x - 3) - 1| - 1)^2 + 4) + 1) - 2) + 3)/(1(1cos(1π(-1(-2|(-1x - 3) - 1| - 1)^2 + 4) + 1) - 2) - 2) negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 51 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-1(2sin(1π(3|((-3x - 1)/(1x - 2)) + 2| - 1) + 3) + 3)^2 - 5(2sin(1π(3|((-3x - 1)/(1x - 2)) + 2| - 1) + 3) + 3) - 3) - 2| - 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 17 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(2|(1(-1((1x - 3)/(-1x - 1)) - 3)^2 + 5(-1((1x - 3)/(-1x - 1)) - 3) + 2) + 1| - 2) - 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) = 2cos(1π(-2|(-5((3(3x - 4) + 2)/(1(3x - 4))) - 3) + 2| - 1) - 1) + 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-3((-2((3(-1sin(2πx) - 1) - 2)/(-2(-1sin(2πx) - 1) + 3)) - 2)/(-2((3(-1sin(2πx) - 1) - 2)/(-2(-1sin(2πx) - 1) + 3)) + 3))^2 - 2((-2((3(-1sin(2πx) - 1) - 2)/(-2(-1sin(2πx) - 1) + 3)) - 2)/(-2((3(-1sin(2πx) - 1) -... | 20 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|(3(-3(-1sin(3π(-1x^2 + 3x - 2)) - 1) - 5)^2 + 2(-3(-1sin(3π(-1x^2 + 3x - 2)) - 1) - 5) - 2)| - 1 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3((-3((2((2(-3cos(3πx - 2) + 1))/(-1(-3cos(3πx - 2) + 1) - 3)) + 3)/(1((2(-3cos(3πx - 2) + 1))/(-1(-3cos(3πx - 2) + 1) - 3)))) + 3)/(2((2((2(-3cos(3πx - 2) + 1))/(-1(-3cos(3πx - 2) + 1) - 3)) + 3)/(1((2(-3cos(3π... | 30 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1|(-3((2(2cos(3π((1x + 2)/(-3x + 1)) + 3) - 2) + 1)/(-2(2cos(3π((1x + 2)/(-3x + 1)) + 3) - 2) + 3)) - 4) - 1| is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 6 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2|((1(-1|((1(3cos(1πx - 2)) + 2)/(2(3cos(1πx - 2)) + 1)) - 3| + 2) + 3)/(-1(-1|((1(3cos(1πx - 2)) + 2)/(2(3cos(1πx - 2)) + 1)) - 3| + 2) + 2)) - 1| + 2 positive?\n\nPresent the answer in LaTex format: \\boxed{Your... | 21 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-1|(-2(-1e^(2(-4x^2 + 5x)) + 1)^2 - 5(-1e^(2(-4x^2 + 5x)) + 1) + 5)| + 3)| - 1 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(5(-3|(1cos(1πx) + 2) - 1| - 3)^2 + 4(-3|(1cos(1πx) + 2) - 1| - 3) - 4) + 2)/(1(5(-3|(1cos(1πx) + 2) - 1| - 3)^2 + 4(-3|(1cos(1πx) + 2) - 1| - 3) - 4) - 3)) - 3| - 3 is strictly decreasing.\n\nPresent t... | 10 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2sin(1π(3|(-2|(-1(1|x - 1| - 3)^2 - 2(1|x - 1| - 3))| + 2) - 1| - 2) - 2) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1(-2sin(3π(-3|(-1(5x^2 + 3x - 4) + 5) + 1| - 3) + 1))^2 - 3(-2sin(3π(-3|(-1(5x^2 + 3x - 4) + 5) + 1| - 3) + 1)) + 5 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2sin(1π(-1(5(2|((-2x - 2)/(-1x + 2)) - 3| - 3)^2 - 2(2|((-2x - 2)/(-1x + 2)) - 3| - 3) + 1)^2 + 2(5(2|((-2x - 2)/(-1x + 2)) - 3| - 3)^2 - 2(2|((-2x - 2)/(-1x + 2)) - 3| - 3) + 1) + 5) + 1) - 2 negative?\n\nPresent... | 12 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (2(1|(-2sin(3π(-3(1|x + 2| - 3) - 1)) + 1) - 2| - 3) + 2)/(2(1|(-2sin(3π(-3(1|x + 2| - 3) - 1)) + 1) - 2| - 3) + 3) negative?\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) = 2|(1|(-2|(-2cos(3π(-2x^2 + 1x + 4) + 3) + 3) - 1| + 2) + 1| - 2) - 2| - 1 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "For the function f(x) = 1|((1(-3sin(1π((1((-3x - 2)/(1x + 3)) + 2)/(-1((-3x - 2)/(1x + 3)) + 2))) + 1) - 3)/(3(-3sin(1π((1((-3x - 2)/(1x + 3)) + 2)/(-1((-3x - 2)/(1x + 3)) + 2))) + 1) + 1)) - 3| - 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer... | 6 | MATH |
[
{
"content": "For the function f(x) = 3|(1sin(3π((1((-1(2|x + 3| - 3) - 2)/(-1(2|x + 3| - 3) - 3)) - 3)/(2((-1(2|x + 3| - 3) - 2)/(-1(2|x + 3| - 3) - 3)))) + 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}... | 9 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (-3((-2(-2sin(3π(2|(4x^2 + 3x + 3) + 2| + 1) - 3) - 1) + 2)/(-3(-2sin(3π(2|(4x^2 + 3x + 3) + 2| + 1) - 3) - 1) - 2)) - 1)/(3((-2(-2sin(3π(2|(4x^2 + 3x + 3) + 2| + 1) - 3) - 1) + 2)/(-3(-2sin(3π(2|(4x^2 + 3x ... | 40 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(3(3(1cos(2π(3x - 4) - 1) - 3) - 1)^2 + 4(3(1cos(2π(3x - 4) - 1) - 3) - 1) - 1) + 1 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) = 3(1|(-3cos(3π(3|(4x^2 - 3) + 3| - 3) + 2)) + 3| + 3)^2 - 5(1|(-3cos(3π(3|(4x^2 - 3) + 3| - 3) + 2)) + 3| + 3) + 2 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (3(4(-1|(1(-2|x| + 2)^2 + 1(-2|x| + 2) - 1) - 1| - 2)^2 + 5(-1|(1(-2|x| + 2)^2 + 1(-2|x| + 2) - 1) - 1| - 2) - 2) - 3)/(2(4(-1|(1(-2|x| + 2)^2 + 1(-2|x| + 2) - 1) - 1| - 2)^2 + 5(-1|(1(-2|x| + 2)^2 + 1(-2|x|... | 4 | MATH |
[
{
"content": "For the function f(x) = -1(3|((1((2(2x^2 - 5x - 3) - 2)/(3(2x^2 - 5x - 3) + 2)) + 1)/(2((2(2x^2 - 5x - 3) - 2)/(3(2x^2 - 5x - 3) + 2)) - 1))| - 1)^2 + 3(3|((1((2(2x^2 - 5x - 3) - 2)/(3(2x^2 - 5x - 3) + 2)) + 1)/(2((2(2x^2 - 5x - 3) - 2)/(3(2x^2 - 5x - 3) + 2)) - 1))| - 1) + 4, find the number of m... | 5 | MATH |
[
{
"content": "For the function f(x) = (-3((3((-1(1(3cos(3πx) - 3)^2 + 4(3cos(3πx) - 3) - 1) + 3)/(-2(1(3cos(3πx) - 3)^2 + 4(3cos(3πx) - 3) - 1) + 3)) - 1)/(2((-1(1(3cos(3πx) - 3)^2 + 4(3cos(3πx) - 3) - 1) + 3)/(-2(1(3cos(3πx) - 3)^2 + 4(3cos(3πx) - 3) - 1) + 3)) + 3)) - 3)/(-2((3((-1(1(3cos(3πx) - 3)^2 + 4(3cos... | 60 | MATH |
[
{
"content": "For the function f(x) = -3|((-1(-1(3|(1x^2 + 3x - 2) + 3|)^2 + 1) - 2)/(-1(-1(3|(1x^2 + 3x - 2) + 3|)^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"
}
] | 4 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -4(-1|(-3(-5(-1sin(2πx - 1))^2 - 4) - 5) - 3|) + 3 negative?\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) = (-1(-2cos(2π(-4(-3|(2|x + 2| + 2) + 2|)^2 + 3(-3|(2|x + 2| + 2) + 2|) + 5)) + 2) - 1)/(-2(-2cos(2π(-4(-3|(2|x + 2| + 2) + 2|)^2 + 3(-3|(2|x + 2| + 2) + 2|) + 5)) + 2) + 3) is strictly increasing.\n\nPresent ... | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = (2(-1(2(-2cos(2π((-3x + 2)/(1x + 2)) - 3)) + 3) + 3))/(-2(-1(2(-2cos(2π((-3x + 2)/(1x + 2)) - 3)) + 3) + 3) + 1) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 19 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 1cos(3π(-4((-1(4(3|x - 1| - 2) - 1))/(2(4(3|x - 1| - 2) - 1) - 1)) - 4) - 3) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = -2sin(3π(4((1(1(-3|x - 1| + 2) + 5) - 3)/(-1(1(-3|x - 1| + 2) + 5) - 2))^2 - 4((1(1(-3|x - 1| + 2) + 5) - 3)/(-1(1(-3|x - 1| + 2) + 5) - 2)) - 5)) + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxe... | 40 | MATH |
[
{
"content": "For the function f(x) = -2sin(3π(4((-1(-2((-2x - 1)/(-1x + 2)) - 2) + 3)/(-1(-2((-2x - 1)/(-1x + 2)) - 2) - 3))^2 + 1((-1(-2((-2x - 1)/(-1x + 2)) - 2) + 3)/(-1(-2((-2x - 1)/(-1x + 2)) - 2) - 3)) + 4) - 3) + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent ... | 42 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 4(-1sin(2π(-3(1((1x)/(-1x - 3))^2 + 3((1x)/(-1x - 3)) + 1) - 3)) + 2) - 2 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) = (-1(2|(1(2cos(2π(-4x + 5) - 1) - 2) + 4) + 3| + 3) - 2)/(-1(2|(1(2cos(2π(-4x + 5) - 1) - 2) + 4) + 3| + 3) + 2) is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 80 | MATH |
[
{
"content": "For the function f(x) = -1cos(3π(-1(4(-4(-3x - 4)^2 - 5(-3x - 4) - 2) - 3) - 5) + 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"
}
] | 5 | MATH |
[
{
"content": "For the function f(x) = 3|(3((-1(-3(3x^2 - 1x - 2)) - 3)/(2(-3(3x^2 - 1x - 2))))^2 + 5((-1(-3(3x^2 - 1x - 2)) - 3)/(2(-3(3x^2 - 1x - 2)))) + 4) - 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}",
"... | 4 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(2π((-2(-2((2(-3x^2 - 5x) - 1)/(1(-3x^2 - 5x) - 2))^2 - 4((2(-3x^2 - 5x) - 1)/(1(-3x^2 - 5x) - 2)) - 3) - 3)/(3(-2((2(-3x^2 - 5x) - 1)/(1(-3x^2 - 5x) - 2))^2 - 4((2(-3x^2 - 5x) - 1)/(1(-3x^2 - 5x) - 2)) ... | 4 | MATH |
[
{
"content": "For the function f(x) = -2sin(1π(-2|(1((3(-3x + 3) + 1)/(1(-3x + 3) - 2))^2 + 2((3(-3x + 3) + 1)/(1(-3x + 3) - 2)) + 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"
}
... | 24 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-4(3cos(2π(1((3x - 1)/(-1x - 1)) - 4) + 3) + 1) - 1) - 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "For the function f(x) = -3(3|(-3cos(2π(-5((-2x - 3)/(-1x + 3)) - 2) + 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"
}
] | 23 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(-2sin(3π((-1(5((-1x + 2)/(1x + 1))^2 + 2((-1x + 2)/(1x + 1)) + 4) - 1)/(-1(5((-1x + 2)/(1x + 1))^2 + 2((-1x + 2)/(1x + 1)) + 4) + 3)) + 3) + 2)^2 - 4(-2sin(3π((-1(5((-1x + 2)/(1x + 1))^2 + 2((-1x + 2)/(1x + 1)) ... | 23 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 2(3((2(-3cos(2π((-2x - 1)/(1x + 2)) + 1)) - 2)/(1(-3cos(2π((-2x - 1)/(1x + 2)) + 1)) - 2)) - 3)^2 + 1(3((2(-3cos(2π((-2x - 1)/(1x + 2)) + 1)) - 2)/(1(-3cos(2π((-2x - 1)/(1x + 2)) + 1)) - 2)) - 3) - 4 negative?\n\n... | 11 | MATH |
[
{
"content": "For the function f(x) = -2|(-3(5(-2|(-3|x + 2| + 3) - 1| + 3) - 5) + 5)| + 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) = -4(-5(-1((1(-1|x - 2| + 2) + 3)/(-2(-1|x - 2| + 2) + 1))^2 + 3)^2 + 5(-1((1(-1|x - 2| + 2) + 3)/(-2(-1|x - 2| + 2) + 1))^2 + 3) - 5) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "For the function f(x) = -1(-2|(5(-1(2sin(1πx + 2)) + 1)^2 - 1) - 3| - 3)^2 - 5(-2|(5(-1(2sin(1πx + 2)) + 1)^2 - 1) - 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"
}
] | 33 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3|(-2|((-2(-1cos(3π(2x^2 - 2x + 4)) - 3) + 1)/(3(-1cos(3π(2x^2 - 2x + 4)) - 3))) - 1| + 2)| + 2 negative?\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) = -1sin(1π(-2((-1(1|((-3x - 2)/(2x)) - 1| - 1) + 2)/(2(1|((-3x - 2)/(2x)) - 1| - 1) - 1)) - 4) - 3) - 2 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -2(1cos(1π(4((3(1x - 3) - 1)/(-2(1x - 3) - 1))^2 - 3((3(1x - 3) - 1)/(-2(1x - 3) - 1)) + 5) + 1) + 1) - 4 is strictly decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 16 | MATH |
[
{
"content": "For the function f(x) = (3(-1(-3cos(1π((-1(1x^2 - 2x - 3) - 1)/(-2(1x^2 - 2x - 3) + 3))) + 2)^2 - 3(-3cos(1π((-1(1x^2 - 2x - 3) - 1)/(-2(1x^2 - 2x - 3) + 3))) + 2) + 2) + 1)/(2(-1(-3cos(1π((-1(1x^2 - 2x - 3) - 1)/(-2(1x^2 - 2x - 3) + 3))) + 2)^2 - 3(-3cos(1π((-1(1x^2 - 2x - 3) - 1)/(-2(1x^2 - 2x -... | 7 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2(-3cos(1π(-1(1|((1x + 2)/(-2x + 1)) + 2| + 3) - 2) + 1) + 3)^2 + 5(-3cos(1π(-1(1|((1x + 2)/(-2x + 1)) + 2| + 3) - 2) + 1) + 3) + 5 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your... | 6 | MATH |
[
{
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1(-3|(5(-1|(-3sin(3πx + 2) - 1)| + 2) - 2) + 3| + 3) + 2 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3(-3|(1|(2((3x - 1)/(2x + 2))^2 - 4((3x - 1)/(2x + 2))) - 2| + 1) - 3| - 1) + 4 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) = -2cos(3π(4((3(-5(-1x + 5) - 4) - 3)/(1(-5(-1x + 5) - 4) - 2)) - 2) - 1) - 3 positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "For the function f(x) = 3(5(1|(-3cos(2π(-4x^2 + 2) - 1)) - 2| - 1)^2 + 2(1|(-3cos(2π(-4x^2 + 2) - 1)) - 2| - 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"
}
] | 14 | MATH |
[
{
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -3((2(2sin(1π(2|((2x)/(2x - 3)) - 1| + 2) - 3) - 2) + 2)/(1(2sin(1π(2|((2x)/(2x - 3)) - 1| + 2) - 3) - 2) - 3)) - 4 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
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