17. given the below functions of f(x), g(x), and h(x), determine which of them has the greatest rate of…

Explanation:

Step1: Recall rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 4$ and $b = 6$.

Step2: Find rate of change of $f(x)$

From the graph of $f(x)$, when $x = 4$, $f(4)\approx - 4$ and when $x = 6$, $f(6)=16$. The rate of change of $f(x)$ is $\frac{f(6)-f(4)}{6 - 4}=\frac{16-(-4)}{2}=\frac{20}{2}=10$.

Step3: Find rate of change of $g(x)$

From the table of $g(x)$, when $x = 4$, $g(4)=19$ and when $x = 6$, $g(6)=35$. The rate of change of $g(x)$ is $\frac{g(6)-g(4)}{6 - 4}=\frac{35 - 19}{2}=\frac{16}{2}=8$.

Step4: Find rate of change of $h(x)$

First, find $h(4)$ and $h(6)$. $h(x)=-x^{2}+8x + 12$. $h(4)=-4^{2}+8\times4 + 12=-16 + 32+12=28$. $h(6)=-6^{2}+8\times6 + 12=-36+48 + 12=24$. The rate of change of $h(x)$ is $\frac{h(6)-h(4)}{6 - 4}=\frac{24 - 28}{2}=\frac{-4}{2}=-2$.

Answer:

$f(x)$ has the greatest rate of change on the interval $4\leq x\leq6$.