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 a. x= (type an exact answer, using π as needed. use a comma to separate answers as needed.) b. there is no local maximum.

Explanation:

Step1: Recall local - maximum definition

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

Step2: Analyze the graph

From the given graph of the function \(y = f(x)\), we can see that the function has a peak at the point \((\frac{\pi}{2},9)\). In an open - interval around \(x=\frac{\pi}{2}\), the value of \(y = f(x)\) is \(9\) and for other \(x\) values in that open interval, \(f(x)<9\).

Step3: Recall local - minimum definition

A function \(y = f(x)\) has a local minimum at \(x = c\) if \(f(c)\leq f(x)\) for all \(x\) in some open interval containing \(c\).

Step4: Analyze the graph for local minimum

From the graph, the function has a trough at the point \((-\frac{\pi}{2}, - 9)\). In an open - interval around \(x =-\frac{\pi}{2}\), the value of \(y = f(x)\) is \(-9\) and for other \(x\) values in that open interval, \(f(x)>-9\).

Answer:

(a) \(x=\frac{\pi}{2}\), local maximum value is \(9\)
(b) \(x =-\frac{\pi}{2}\), local minimum value is \(-9\)