Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = sin 2x + 3 on [-π , π]

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To find the critical points of the function ƒ(x) = sin(2x) + 3 on the interval [-π, π], first compute the derivative of the function. The derivative of sin(2x) is 2cos(2x), so the derivative of ƒ(x) is ƒ'(x) = 2cos(2x).

Set the derivative ƒ'(x) = 2cos(2x) equal to zero to find the critical points. Solve the equation 2cos(2x) = 0, which simplifies to cos(2x) = 0.

The solutions to cos(2x) = 0 are 2x = π/2 + kπ, where k is an integer. Solve for x to find x = π/4 + kπ/2.

Determine which solutions for x fall within the interval [-π, π]. Substitute k = -2, -1, 0, 1, 2 into x = π/4 + kπ/2 to find the valid critical points within the interval.

Evaluate the function ƒ(x) = sin(2x) + 3 at the critical points and at the endpoints x = -π and x = π to determine the absolute maximum and minimum values. Compare these values to identify the absolute extrema.

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주요 개념

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

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [-π, π] indicates that the function is being analyzed from -π to π, including both endpoints. Understanding interval notation is crucial for correctly identifying where to evaluate the function and its critical points.

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).ƒ(x) = sin 2x + 3 on [-π , π]