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 \(c\).

Step2: Examine the graph

By looking at the graph, we find the peaks. The function \(y = f(x)\) has a relative maximum at \(x = 2\). The value of the function at \(x = 2\) is \(y=4\).

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 \(c\).

Step4: Examine the graph for minima

The function \(y = f(x)\) has relative minima at \(x=-1\) and \(x = 4\). The value of the function at \(x=-1\) is \(y = 1\) and at \(x = 4\) is \(y=1\).

Answer:

(a) The number(s) at which \(f\) has a relative maximum is \(2\). The relative - maximum is \(4\).
(b) The numbers at which \(f\) has a relative minimum are \(-1,4\). The relative - minima are \(1,1\).