Question

(i) identify the critical numbers of ( f ). (enter your answers as a comma - separated list.)( x = -1,1,3 )✔️
(ii) identify the open interval(s) on which ( f ) is increasing or decreasing. (enter your answer using interval notation.)
increasing ( -1,1,3,infty) ) your answer cannot be understood or graded. more information
decreasing ( (-infty, - 1,1,3 )
(iii) determine whether ( f ) has a relative maximum, a relative minimum, or neither at each critical number. (enter your answers as comma - separated lists.)
relative minimum ( x = ) dne
relative maximum ( x = ) dne

Explanation:

Step1: Analyze increasing intervals

Looking at the graph, the function rises when \(x\) moves from \(-1\) to \(1\), and from \(3\) to \(+\infty\). These are open intervals, so:
\((-1, 1), (3, \infty)\)

Step2: Analyze decreasing intervals

The function falls when \(x\) moves from \(-\infty\) to \(-1\), and from \(1\) to \(3\). These are open intervals, so:
\((-\infty, -1), (1, 3)\)

Step3: Identify relative extrema

• At \(x=-1\): The function changes from decreasing to increasing, so this is a relative minimum.
• At \(x=1\): The function changes from increasing to decreasing, so this is a relative maximum.
• At \(x=3\): The function changes from decreasing to increasing, so this is a relative minimum.

Answer:

(ii)

Increasing: \((-1, 1), (3, \infty)\)
Decreasing: \((-\infty, -1), (1, 3)\)

(iii)

relative minimum \(x = -1, 3\)
relative maximum \(x = 1\)