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?
find the local maximum value. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the local maximum value is 11.
(type an exact answer, using π as needed. use a comma to separate answers as needed.)
b. there is no local maximum.
(b) find the number(s) x at which f has a local minimum. select the correct choice 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 minimum.

Explanation:

Step1: Recall the definition of local minimum

A local minimum of a function \(y = f(x)\) is a point \((x_0,f(x_0))\) such that \(f(x_0)\leq f(x)\) for all \(x\) in some open interval containing \(x_0\).

Step2: Analyze the given graph

Looking at the graph, we can see that the function \(f\) has a local minimum at \(x =-\frac{\pi}{2}\) because in the neighborhood of \(x =-\frac{\pi}{2}\), the value of the function \(f(-\frac{\pi}{2})=- 11\) is less than the values of the function for \(x\) values close to \(-\frac{\pi}{2}\).

Answer:

A. \(x =-\frac{\pi}{2}\)