Question

for which interval is the average rate of change of f(x) negative?
from x = -3.5 to x = -1
from x = -3 to x = 3

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. It is negative when $f(b)-f(a)<0$ (i.e., $f(b) a$.

Step2: Analyze the first interval

For the interval $x=-3.5$ to $x = - 1$, from the graph, when $x=-3.5$, $f(-3.5)\approx - 3$ and when $x=-1$, $f(-1)\approx1$. Since $1>-3$, $\frac{f(-1)-f(-3.5)}{-1-(-3.5)}=\frac{f(-1)-f(-3.5)}{2.5}>0$.

Step3: Analyze the second interval

For the interval $x=-3$ to $x = 3$, when $x=-3$, $f(-3)\approx - 2$ and when $x = 3$, $f(3)\approx - 3$. Since $-3<-2$ and $3>-3$, $\frac{f(3)-f(-3)}{3-(-3)}=\frac{f(3)-f(-3)}{6}<0$.

Answer:

from $x=-3$ to $x = 3$