Question

a continuous function f is defined on the closed interval - 5 < x < 6 and is shown in the graph below. for how many values of b, - 5 < b < 6, is the average rate of change of f on the interval b, 5 equal to 0? give a reason for your answer.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. For the interval $[b,5]$, the average rate of change is $\frac{f(5)-f(b)}{5 - b}$. We want this to be equal to 0, so $f(5)-f(b)=0$ (since $5 - b
eq0$ for $b
eq5$), which implies $f(b)=f(5)$.

Step2: Analyze the graph

Find the value of $f(5)$ from the graph. Then, look for the number of times the function $y = f(x)$ takes the value of $f(5)$ for $- 5 < b < 5$.

Step3: Count the number of intersections

By observing the graph, we count the number of times the horizontal line $y = f(5)$ intersects the graph of $y = f(x)$ for $-5 < b < 5$.

Answer:

2