Question

on which interval does the function f(x) = 2x² - x have an average rate of change equal to
3?
x = 0 to x = 2
x = -7 to x = -3
x = 4 to x = 5
x = -5 to x = -2

Explanation:

Step1: Recall average rate of change formula

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

Step2: For the interval $x = 0$ to $x=2$

$f(x)=2x^{2}-x$. First, find $f(0)$ and $f(2)$.
$f(0)=2(0)^{2}-0 = 0$.
$f(2)=2(2)^{2}-2=2\times4 - 2=6$.
The average rate of change is $\frac{f(2)-f(0)}{2 - 0}=\frac{6 - 0}{2}=3$.

Step3: For the interval $x=-7$ to $x = - 3$

$f(-7)=2(-7)^{2}-(-7)=2\times49 + 7=98 + 7=105$.
$f(-3)=2(-3)^{2}-(-3)=2\times9+3 = 18 + 3=21$.
The average rate of change is $\frac{f(-3)-f(-7)}{-3-(-7)}=\frac{21 - 105}{-3 + 7}=\frac{-84}{4}=-21$.

Step4: For the interval $x = 4$ to $x=5$

$f(4)=2(4)^{2}-4=2\times16-4=32 - 4=28$.
$f(5)=2(5)^{2}-5=2\times25-5=50 - 5=45$.
The average rate of change is $\frac{f(5)-f(4)}{5 - 4}=\frac{45 - 28}{1}=17$.

Step5: For the interval $x=-5$ to $x=-2$

$f(-5)=2(-5)^{2}-(-5)=2\times25 + 5=50+5 = 55$.
$f(-2)=2(-2)^{2}-(-2)=2\times4 + 2=8 + 2=10$.
The average rate of change is $\frac{f(-2)-f(-5)}{-2-(-5)}=\frac{10 - 55}{-2 + 5}=\frac{-45}{3}=-15$.

Answer:

We need to check which interval gives the desired average - rate of change value (not given in the problem statement. But the above are the average rates of change for each of the given intervals). If we assume we are looking for an average rate of change value among the calculated ones, we would choose based on that value. For example, if the desired value is 3, the interval is $x = 0$ to $x=2$; if - 21, the interval is $x=-7$ to $x=-3$; if 17, the interval is $x = 4$ to $x=5$; if - 15, the interval is $x=-5$ to $x=-2$.