(a) find the open intervals on which the function shown in the graph is increasing and decreasing. (b)…

(a) find the open intervals on which the function shown in the graph is increasing and decreasing. (b) identify the functions local and absolute extreme values, if any, saying where they occur. (b) if the function has an absolute maximum, where does it occur? select the correct choice below and fill in any answer boxes within your choice. a. an absolute maximum occurs at the point(s) (type an ordered - pair. use a comma to separate answers as needed.) b. the function has no absolute maximum. if the function has other local maxima, where do they occur? since a list of local maxima automatically includes the absolute maximum, do not include the absolute maximum in the list of local maxima. select the correct choice below and fill in any answer boxes within your choice.

(a) find the open intervals on which the function shown in the graph is increasing and decreasing. (b) identify the functions local and absolute extreme values, if any, saying where they occur. (b) if the function has an absolute maximum, where does it occur? select the correct choice below and fill in any answer boxes within your choice. a. an absolute maximum occurs at the point(s) (type an ordered - pair. use a comma to separate answers as needed.) b. the function has no absolute maximum. if the function has other local maxima, where do they occur? since a list of local maxima automatically includes the absolute maximum, do not include the absolute maximum in the list of local maxima. select the correct choice below and fill in any answer boxes within your choice.

Answer

Explanation:

Step1: Recall increasing - decreasing rules

A function is increasing when the graph rises from left - to - right and decreasing when it falls from left - to - right.

Step2: Identify increasing intervals

By observing the graph, the function is increasing on the intervals $(-8,-2)$ and $(2,4)$.

Step3: Identify decreasing intervals

The function is decreasing on the intervals $(-2,2)$ and $(4,8)$.

Step4: Recall extreme - value rules

Local maxima occur where the function changes from increasing to decreasing and local minima occur where the function changes from decreasing to increasing. Absolute maxima and minima are the highest and lowest values of the function over its entire domain.

Step5: Identify local and absolute extreme values

The local maximum occurs at $x=-2$ with $y = 6$ and at $x = 4$ with $y=1$. The local minimum occurs at $x = 2$ with $y=-2$. Since the graph is bounded within the given domain $[-8,8]$, the absolute maximum is $y = 6$ at $x=-2$ and the absolute minimum is $y=-4$ at $x = 8$.

Answer:

(a) Increasing intervals: $(-8,-2),(2,4)$; Decreasing intervals: $(-2,2),(4,8)$ (b) Absolute maximum occurs at the point $(-2,6)$; Local maxima (excluding absolute maximum) occur at $(4,1)$; Local minimum occurs at $(2,-2)$; Absolute minimum occurs at $(8,-4)$