Question

using the given graph of the function f, find the following. (a) the numbers, if any, at which f has a local maximum. what are these local maximum values? (b) the numbers, if any, at which f has a local minimum. what are these local minimum values? (a) find the value(s) of x at which f has a local maximum. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x = □ (type an exact answer, using π as needed. use a comma to separate answers as needed.) b. there is no local maximum value.

Explanation:

Step1: Recall local - maximum and minimum definitions

Local maximum is a point where the function value is greater than or equal to the values at nearby points, and local minimum is a point where the function value is less than or equal to the values at nearby points.

Step2: Analyze the graph

From the graph, we can see that the function has a local maximum at \(x =-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\), and a local minimum at \(x = \pi\) and \(x=-\pi\).

Answer:

(a) A. \(x =-\frac{\pi}{2},\frac{\pi}{2}\)
(b) The numbers at which \(f\) has a local minimum are \(x =-\pi,\pi\)