Question

the graph of ( f ) is given.
(a) on what open intervals is ( f ) increasing?
(b) on what open intervals is ( f ) decreasing?
(c) what are the ( x )-values of the local minima?
(d) what are the ( x )-values of the local maxima?
(a) select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is increasing on the open interval(s)
(type your answer in interval notation. use a comma to separate answers as needed )
b. there are no open intervals on which the function ( f ) is increasing

Explanation:

Step1: Recall the relationship between \(f'\) and \(f\)

If \(f'(x)>0\) on an open interval \(I\), then the function \(y = f(x)\) is increasing on \(I\).

Step2: Analyze the graph of \(f'\)

Looking at the graph of \(y = f'(x)\), we find the intervals where \(y=f'(x)>0\).
From the graph, \(f'(x)>0\) when \(x\in(-\infty, - 1)\cup(3,\infty)\)

Answer:

A. The function \(f\) is increasing on the open interval(s) \((-\infty,-1),(3,\infty)\)