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

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 $\frac{f(b)-f(a)}{b - a}$. If $f(b)-f(a)<0$ and $b - a>0$ (since $b>a$ for an interval $[a,b]$), the average rate of change is negative. This means the function value at the end - point $b$ is less than the function value at the start - point $a$.

Step2: Analyze the first option

For the interval $x=-3.5$ to $x = - 1$: Looking at the graph, $f(-3.5)\approx - 3$ and $f(-1)\approx1$. Then $\frac{f(-1)-f(-3.5)}{-1-(-3.5)}=\frac{1 - (-3)}{-1 + 3.5}=\frac{4}{2.5}>0$.

Step3: Analyze the second option

For the interval $x=-3$ to $x = 3$: $f(-3)\approx - 3$ and $f(3)\approx - 3$. Then $\frac{f(3)-f(-3)}{3-(-3)}=\frac{-3-(-3)}{3 + 3}=\frac{0}{6}=0$.

Step4: Analyze the third option

For the interval $x = 0$ to $x=2.5$: $f(0)\approx1$ and $f(2.5)\approx - 3$. Then $\frac{f(2.5)-f(0)}{2.5 - 0}=\frac{-3 - 1}{2.5}=\frac{-4}{2.5}<0$.

Answer:

from $x = 0$ to $x = 2.5$