Answer
$${f_{avg}} = 2,{\text{ }}x = 8$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^3},{\text{ }}\left[ {0,2} \right] \cr
& {\text{Calculate the average value}} \cr
& {f_{avg}} = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& {f_{avg}} = \frac{1}{{2 - 0}}\int_4^9 {\frac{1}{{\sqrt x }}} dx \cr
& {f_{avg}} = \frac{1}{2}\int_0^2 {{x^3}} dx \cr
& {f_{avg}} = \frac{1}{2}\left[ {\frac{1}{4}{x^4}} \right]_0^2 \cr
& {f_{avg}} = \frac{1}{8}\left[ {{x^4}} \right]_0^2 \cr
& {f_{avg}} = \frac{1}{8}\left[ {{{\left( 2 \right)}^4} - {{\left( 0 \right)}^4}} \right] \cr
& {f_{avg}} = 2 \cr
& \cr
& {\text{Let }}{f_{avg}} = f\left( x \right) \cr
& 2 = {x^3} \cr
& x = 8 \cr
& \cr
& {f_{avg}} = 2,{\text{ }}x = 8 \cr} $$
