choose all that give the correct average rate of change for the specified function on the specified…

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 $\frac{f(b)-f(a)}{b - a}$.

Step2: Check option A

For $f(x)=x^{2}+2$ on $[1,4]$, $f(4)=4^{2}+2=16 + 2=18$, $f(1)=1^{2}+2=3$. Then $\frac{f(4)-f(1)}{4 - 1}=\frac{18 - 3}{3}=\frac{15}{3}=5$.

Step3: Check option B

For $f(x)=2x^{2}-2$ on $[1,3]$, $f(3)=2\times3^{2}-2=2\times9 - 2=16$, $f(1)=2\times1^{2}-2=0$. Then $\frac{f(3)-f(1)}{3 - 1}=\frac{16-0}{2}=8\neq4$.

Step4: Check option C

For $f(x)=4x^{2}+2$ on $[0,3]$, $f(3)=4\times3^{2}+2=4\times9 + 2=38$, $f(0)=4\times0^{2}+2=2$. Then $\frac{f(3)-f(0)}{3 - 0}=\frac{38 - 2}{3}=\frac{36}{3}=12\neq6$.

Step5: Check option D

For $f(x)=3x^{2}-1$ on $[2,5]$, $f(5)=3\times5^{2}-1=3\times25 - 1=74$, $f(2)=3\times2^{2}-1=3\times4 - 1=11$. Then $\frac{f(5)-f(2)}{5 - 2}=\frac{74 - 11}{3}=\frac{63}{3}=21$.

Answer:

A. The average rate of change of $f(x)=x^{2}+2$ on $[1,4]$ is 5. D. The average rate of change of $f(x)=3x^{2}-1$ on $[2,5]$ is 21.