compare the average rate of change for f(x)=∛x and g(x)=∛x + 5 for 0≤x≤4. select the correct choice below…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Here, $a = 0$ and $b = 4$.

Step2: Calculate average rate of change of $f(x)=\sqrt[3]{x}$

$f(4)=\sqrt[3]{4}$, $f(0)=\sqrt[3]{0}=0$. The average rate of change of $f(x)$ is $\frac{f(4)-f(0)}{4 - 0}=\frac{\sqrt[3]{4}-0}{4}=\frac{\sqrt[3]{4}}{4}\approx\frac{1.587}{4}=0.397$.

Step3: Calculate average rate of change of $g(x)=\sqrt[3]{x}+5$

$g(4)=\sqrt[3]{4}+5$, $g(0)=\sqrt[3]{0}+5 = 5$. The average rate of change of $g(x)$ is $\frac{g(4)-g(0)}{4 - 0}=\frac{(\sqrt[3]{4}+5)-5}{4}=\frac{\sqrt[3]{4}}{4}\approx0.397$.

Answer:

C. The average rates of change of $f(x)$ and $g(x)$ are the same, $0.397$.