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 number(s) x at which f has a local maximum. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. x = \\( \frac { \pi } { 2 } \\)
(type an exact answer, using \\( \pi \\) as needed. use a comma to separate answers as needed.)
b. there is no local maximum.
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
(type an exact answer, using \\( \pi \\) as needed. use a comma to separate answers as needed.)
b. there is no local maximum.

Explanation:

Step1: Recall the definition of local maximum

A local maximum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\).

Step2: Analyze the graph

Looking at the given graph of the function \(f\), we can see that at \(x=\frac{\pi}{2}\), the function has a peak.

Step3: Determine the local maximum value

The \(y -\)coordinate of the point \((\frac{\pi}{2},11)\) gives the local maximum value.

Answer:

A. The local maximum value is \(11\)