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: Analyze the graph

By observing the given graph, we look for the peaks. The function \(y = f(x)\) has a relative maximum at \(x = 3\). The value of the relative - maximum is \(f(3)=3\).

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 \(c\).

Step4: Analyze the graph for relative minimum

By observing the graph, the function has relative minima at \(x = 2\) and \(x = 4\). The values of the relative minima are \(f(2)=1\) and \(f(4)=1\).

Answer:

(a) The number(s) at which \(f\) has a relative maximum is \(3\). The relative maximum is \(3\).
(b) The numbers at which \(f\) has a relative minimum are \(2,4\). The relative minima are \(1,1\).