Answer
The average rate of change of $f\left( x \right)={{x}^{2}}-4x$ from ${{x}_{1}}=5$ to ${{x}_{2}}=9$ is $10$.
Work Step by Step
Consider the given function:
$f\left( x \right)={{x}^{2}}-4x$
The value of the above function at ${{x}_{1}}=5$ is as follows:
$\begin{align}
& f\left( 5 \right)={{5}^{2}}-4\left( 5 \right) \\
& =25-20 \\
& =5
\end{align}$
And
$\begin{align}
& f\left( 9 \right)={{9}^{2}}-4\left( 9 \right) \\
& =81-36 \\
& =45
\end{align}$
So,
The average rate of change is as follows:
$\begin{align}
& \frac{f\left( 9 \right)-f\left( 5 \right)}{9-5}=\frac{45-5}{4} \\
& =\frac{40}{4} \\
& =10
\end{align}$
Therefore, the average rate of change of $f\left( x \right)={{x}^{2}}-4x$ from ${{x}_{1}}=5$ to ${{x}_{2}}=9$ is $10$.
