Question

on which interval does the function $f(x) = 2x^2 - x$ have an average rate of change equal to 3?$x = -7$ to $x = -3LXB0x = 0$ to $x = 2$$x = 4$ to $x = 5$submit

Explanation:

Step1: Recall average rate of change formula

The average rate of change of $f(x)$ from $x=a$ to $x=b$ is $\frac{f(b)-f(a)}{b-a}$. We set this equal to 3.

Step2: Test interval $x=-7$ to $x=-3$

First calculate $f(-7)=2(-7)^2 - (-7)=2(49)+7=98+7=105$, $f(-3)=2(-3)^2 - (-3)=18+3=21$.
$\frac{f(-3)-f(-7)}{-3-(-7)}=\frac{21-105}{4}=\frac{-84}{4}=-21
eq3$

Step3: Test interval $x=-5$ to $x=-2$

Calculate $f(-5)=2(-5)^2 - (-5)=50+5=55$, $f(-2)=2(-2)^2 - (-2)=8+2=10$.
$\frac{f(-2)-f(-5)}{-2-(-5)}=\frac{10-55}{3}=\frac{-45}{3}=-15
eq3$

Step4: Test interval $x=0$ to $x=2$

Calculate $f(0)=2(0)^2 - 0=0$, $f(2)=2(2)^2 - 2=8-2=6$.
$\frac{f(2)-f(0)}{2-0}=\frac{6-0}{2}=3$

Step5: Verify final interval (optional)

Calculate $f(4)=2(4)^2 - 4=32-4=28$, $f(5)=2(5)^2 - 5=50-5=45$.
$\frac{f(5)-f(4)}{5-4}=\frac{45-28}{1}=17
eq3$

Answer:

x = 0 to x = 2