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?

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: Identify local - maximum from the graph

From the graph, we can see that at \(x=\frac{\pi}{2}\), the function \(f(x)\) has a local maximum. The value of the function at \(x = \frac{\pi}{2}\) is \(y = 2\).

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: Identify local - minimum from the graph

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

Answer:

(a) The number at which \(f\) has a local maximum is \(x = \frac{\pi}{2}\), and the local - maximum value is \(2\).
(b) The number at which \(f\) has a local minimum is \(x=-\frac{\pi}{2}\), and the local - minimum value is \(-2\).