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 1. (type an integer or a decimal. use a comma to separate answers as needed.) the relative maximum/maxima 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 \(c\). By observing the graph, we look for the "peaks".

Step2: Identify relative - maximum points

From the graph, the function \(f(x)\) has a relative maximum at \(x = 1\). To find the relative - maximum value, we look at the \(y\) - coordinate of the point. The \(y\) - coordinate of the point on the graph at \(x = 1\) is \(3\).

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 \(c\). By observing the graph, we look for the "valleys".

Step4: Identify relative - minimum points

The function \(f(x)\) has relative minima at \(x=-1\) and \(x = 3\). The \(y\) - coordinate of the point at \(x=-1\) and \(x = 3\) is \(0\).

Answer:

(a) The number(s) at which \(f\) has a relative maximum is/are \(1\). The relative maximum/maxima is/are \(3\).
(b) The number(s) at which \(f\) has a relative minimum is/are \(-1,3\). The relative minimum/minima is/are \(0\).