Question

use the graph to determine the following. (a) find the numbers at which f has a relative maximum. what are these relative maxima? (b) find the numbers at which f has a relative minimum. what are these relative minima? (a) the number(s) at which f has a relative maximum is/are . (type an integer or a decimal. use a comma to separate answers as needed.)

Explanation:

Step1: Recall relative - maximum definition

A function \(y = f(x)\) has a relative maximum at \(x = c\) if \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\). Looking at the graph, we observe the peaks.

Step2: Identify relative - maximum points

From the graph, the function \(f(x)\) has a relative maximum at \(x=- 1\). The value of the function at \(x = - 1\) is \(f(-1)=4\).

Step3: Recall relative - minimum definition

A function \(y = f(x)\) has a relative minimum at \(x = c\) if \(f(c)\leq f(x)\) for all \(x\) in some open interval containing \(c\). Looking at the graph, we observe the valleys.

Step4: Identify relative - minimum points

From the graph, the function \(f(x)\) has relative minima at \(x=-3\) and \(x = 1\). The values of the function at these points are \(f(-3)=1\) and \(f(1)=1\).

Answer:

(a) The number(s) at which \(f\) has a relative maximum is/are \(-1\). The relative maximum is \(4\).
(b) The numbers at which \(f\) has a relative minimum are \(-3,1\). The relative minima are \(1,1\).