Question

use the graph to find: (a) the numbers, if any, at which f has a local maximum. what are these local maxima? (b) the numbers, if any, at which f has a local minimum. what are these local minima? (a) select the correct choice below and fill in any answer boxes within your choice. a. the value(s) of x at which f has a local maximum is/are x = (type an integer. use a comma to separate answers as needed.) b. there are no values of x at which f has a local maximum. (b) select the correct choice below and fill in any answer boxes within your choice. a. the value(s) of x at which f has a local minimum is/are x = (type an integer. use a comma to separate answers as needed.) b. there are no local minima.

Explanation:

Step1: Recall local - maximum definition

A local maximum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\). Looking at the graph, the function \(y = f(x)\) has a local maximum at \(x=- 2\) since the function value at \(x = - 2\) is higher than the function values at nearby points. The value of \(y\) at \(x=-2\) is \(y = 3\).

Step2: Recall local - minimum definition

A local minimum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\leq f(x)\) for all \(x\) in some open interval containing \(c\). The function \(y = f(x)\) has a local minimum at \(x = 0\) since the function value at \(x = 0\) is lower than the function values at nearby points. The value of \(y\) at \(x = 0\) is \(y = 1\).

Answer:

(a) A. The value(s) of \(x\) at which \(f\) has a local maximum is/are \(x=-2\).
(b) A. The value(s) of \(x\) at which \(f\) has a local minimum is/are \(x = 0\).