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 from x = 0 to x = 2.5

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 $f(b)-f(a)<0$ (assuming $b > a$), i.e., when $f(b)

Step2: Analyze option 1

For the interval $x=-3.5$ to $x = - 1$, from the graph, $f(-3.5)<0$ and $f(-1)>0$, so $\frac{f(-1)-f(-3.5)}{-1-(-3.5)}=\frac{f(-1)-f(-3.5)}{2.5}>0$.

Step3: Analyze option 2

For the interval $x=-3$ to $x = 3$, $f(-3)<0$ and $f(3)<0$, but by observing the graph, $f(-3)0$.

Step4: Analyze option 3

For the interval $x = 0$ to $x=2.5$, from the graph, $f(0)>0$ and $f(2.5)<0$. Then $\frac{f(2.5)-f(0)}{2.5 - 0}=\frac{f(2.5)-f(0)}{2.5}<0$.

Answer:

From $x = 0$ to $x=2.5$