Answer
\[x=2\]
Work Step by Step
\[\begin{align}
& f\left( x \right)=8-2x\text{ on }\left[ 0,4 \right] \\
& \text{The average is given by} \\
& {{f}_{avg}}=\frac{1}{b-a}\int_{a}^{b}{f\left( x \right)dx} \\
& \text{Therefore,} \\
& {{f}_{avg}}=\frac{1}{4-0}\int_{0}^{4}{\left( 8-2x \right)dx} \\
& {{f}_{avg}}=\frac{1}{4}\int_{0}^{4}{\left( 8-2x \right)dx} \\
& \text{Integrating} \\
& {{f}_{avg}}=\frac{1}{4}\left[ 8x-{{x}^{2}} \right]_{0}^{4} \\
& {{f}_{avg}}=\frac{1}{4}\left[ 8\left( 4 \right)-{{\left( 4 \right)}^{2}} \right]-\frac{1}{4}\left[ 8\left( 0 \right)-{{\left( 0 \right)}^{2}} \right] \\
& {{f}_{avg}}=\frac{1}{4}\left[ 16 \right]-\frac{1}{4}\left[ 0 \right] \\
& {{f}_{avg}}=4 \\
& \text{The point at which the given function equals its average value is} \\
& 8-2x=4 \\
& -2x=-4 \\
& x=2 \\
\end{align}\]
