Question

on which interval does the function f(x)=-5x^{2}+4x have an average rate of change equal to -26? x = 1 to x = 3 x = 2 to x = 4 x=-2 to x=-1 x = 0 to x = 3

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: Test the first - interval $x = 1$ to $x = 3$

Let $a = 1$ and $b = 3$. First, find $f(1)$ and $f(3)$.
$f(1)=-5\times(1)^2 + 4\times1=-5 + 4=-1$.
$f(3)=-5\times(3)^2+4\times3=-45 + 12=-33$.
The average rate of change is $\frac{f(3)-f(1)}{3 - 1}=\frac{-33-(-1)}{2}=\frac{-33 + 1}{2}=\frac{-32}{2}=-16$.

Step3: Test the second - interval $x=-2$ to $x = - 1$

Let $a=-2$ and $b=-1$.
$f(-2)=-5\times(-2)^2+4\times(-2)=-20-8=-28$.
$f(-1)=-5\times(-1)^2+4\times(-1)=-5 - 4=-9$.
The average rate of change is $\frac{f(-1)-f(-2)}{-1-(-2)}=\frac{-9-(-28)}{-1 + 2}=\frac{-9 + 28}{1}=19$.

Step4: Test the third - interval $x = 2$ to $x = 4$

Let $a = 2$ and $b = 4$.
$f(2)=-5\times(2)^2+4\times2=-20 + 8=-12$.
$f(4)=-5\times(4)^2+4\times4=-80+16=-64$.
The average rate of change is $\frac{f(4)-f(2)}{4 - 2}=\frac{-64-(-12)}{2}=\frac{-64 + 12}{2}=\frac{-52}{2}=-26$.

Step5: Test the fourth - interval $x = 0$ to $x = 3$

Let $a = 0$ and $b = 3$.
$f(0)=-5\times(0)^2+4\times0 = 0$.
$f(3)=-5\times(3)^2+4\times3=-45 + 12=-33$.
The average rate of change is $\frac{f(3)-f(0)}{3 - 0}=\frac{-33-0}{3}=-11$.

Answer:

$x = 2$ to $x = 4$