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 π 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 function \(y = f(x)\) has a local maximum at a point \(x = a\) if \(f(a)\geq f(x)\) for all \(x\) in some open interval containing \(a\). Geometrically, it is a "peak" on the graph of the function.

Step2: Analyze the given graph

Looking at the graph of the function \(y = f(x)\), we can observe the point \((\frac{\pi}{2},11)\). For values of \(x\) in an open interval around \(x=\frac{\pi}{2}\), the function value \(f(x)\) at \(x = \frac{\pi}{2}\) is greater than the function values \(f(x)\) for other \(x\) in that open - interval.

Answer:

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