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 = )
(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.
(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.

Explanation:

Step1: Recall the definition of local maximum

A local maximum of a function \(y = f(x)\) occurs at a point \(x = a\) if \(f(a)\geq f(x)\) for all \(x\) in some open interval containing \(a\). From the graph, we look for the "peaks".
The point \((\frac{\pi}{2},8)\) is a peak. So \(x=\frac{\pi}{2}\) is the \(x -\) value of the local maximum.

Step2: Determine the local maximum value

The \(y -\) value of the local maximum is the value of the function at the \(x -\) value of the local maximum. At \(x = \frac{\pi}{2}\), \(y=f(\frac{\pi}{2}) = 8\)

Step3: Recall the definition of local minimum

A local minimum of a function \(y = f(x)\) occurs at a point \(x = b\) if \(f(b)\leq f(x)\) for all \(x\) in some open interval containing \(b\). From the graph, the point \((-\frac{\pi}{2},-8)\) is a "valley". So \(x =-\frac{\pi}{2}\) is the \(x -\) value of the local minimum.

Step4: Determine the local minimum value

The \(y -\) value of the local minimum is the value of the function at the \(x -\) value of the local minimum. At \(x=-\frac{\pi}{2}\), \(y = f(-\frac{\pi}{2})=-8\)

Answer:

(a)

• For the \(x -\) value of local maximum: A. \(x=\frac{\pi}{2}\)
• For the local maximum value: A. The local maximum value is \(8\)

(b)

• For the \(x -\) value of local minimum: A. \(x =-\frac{\pi}{2}\)
• For the local minimum value: A. The local minimum value is \(-8\)