using the given graph of the function f, find the following. (a) the numbers, if any, at which f has a local…

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).