Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = cos² x on [-π,π]

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First, understand that to determine where the function f(x) = cos²(x) is increasing or decreasing, we need to find its derivative, f'(x).

Calculate the derivative of f(x) = cos²(x) using the chain rule. The derivative of cos²(x) is f'(x) = 2cos(x)(-sin(x)) = -2cos(x)sin(x).

Simplify the expression for the derivative: f'(x) = -sin(2x), using the double angle identity for sine.

Determine the critical points by setting the derivative equal to zero: -sin(2x) = 0. Solve for x to find the critical points within the interval [-π, π].

Analyze the sign of f'(x) = -sin(2x) in each interval determined by the critical points to identify where the function is increasing (f'(x) > 0) and where it is decreasing (f'(x) < 0).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing. In this context, finding the derivative of f(x) = cos² x will help identify the intervals of increase and decrease.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. By analyzing the critical points of f(x) = cos² x within the interval [-π, π], we can establish the behavior of the function in those regions.

First Derivative Test

The First Derivative Test is a method used to determine the nature of critical points by examining the sign of the derivative before and after each critical point. If the derivative changes from positive to negative at a critical point, the function has a local maximum; if it changes from negative to positive, it has a local minimum. This test will help us confirm the intervals where f(x) is increasing or decreasing.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = cos² x on [-π,π]