Question

1. identify the critical points, increasing, and decreasing

Explanation:

Step1: Define critical points

Critical points are where $f^\prime(x)=0$ or $f^\prime(x)$ is undefined. From the graph, slopes are 0 at $x = - 1$ and $x=1$.

Step2: Determine increasing intervals

Function is increasing where slope is positive. From graph, slope is positive for $x\in(-\infty,-1)\cup(1,\infty)$.

Step3: Determine decreasing interval

Function is decreasing where slope is negative. From graph, slope is negative for $x\in(-1,1)$.

Step4: Identify relative extrema

Relative maximum occurs where function changes from increasing to decreasing, at $(-1,3)$. Relative minimum occurs where function changes from decreasing to increasing, at $(1,-3)$.

Step5: Analyze absolute extrema

Since graph extends to $-\infty$ and $\infty$ in $y -$ direction, there are no absolute maximum or minimum.

Answer:

• Increasing: $(-\infty,-1)\cup(1,\infty)$
• Relative minimum: $(1, - 3)$
• Decreasing: $(-1,1)$
• Relative maximum: $(-1,3)$