classwork (tear - out page): use the graph of f given below to answer each of the following. *1. a. on what…

Explanation:

Step1: Determine increasing intervals

A function is increasing when the graph goes up from left - to - right. From the graph, $f$ is increasing on the intervals $(-1,2)$ and $(3,4)$.

Step2: Determine decreasing intervals

A function is decreasing when the graph goes down from left - to - right. From the graph, $f$ is decreasing on the intervals $(-4,-1)$ and $(2,3)$.

Step3: Identify local maximum

A local maximum occurs where the function changes from increasing to decreasing. The local maximum value is $y = 3$ at $x = 2$.

Step4: Identify local minimum

A local minimum occurs where the function changes from decreasing to increasing. The local minimum values are $y = 0$ at $x = 0$ and $y = 2$ at $x = 3$.

Step5: Identify absolute maximum and minimum

Since the graph is only shown on a limited domain, and we don't know the behavior outside of it, the absolute minimum value is $y = 0$ at $x = 0$. There is no absolute maximum shown as the graph continues upwards on the right - hand side.

Step6: Calculate average rate of change

The formula for the average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 2$, $f(-1)=1$ and $f(2)=3$. So, $\frac{f(2)-f(-1)}{2-(-1)}=\frac{3 - 1}{2+1}=\frac{2}{3}$.

Answer:

a. $(-1,2)$ and $(3,4)$ b. $(-4,-1)$ and $(2,3)$ c. $3$ (at $x = 2$) d. $0$ (at $x = 0$) and $2$ (at $x = 3$) e. Absolute minimum is $0$ (at $x = 0$), no absolute maximum shown f. $\frac{2}{3}$