Answer
$${f_{avg}} = \frac{2}{5},{\text{ }}x = \frac{{25}}{4}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{1}{{\sqrt x }},{\text{ }}\left[ {4,9} \right] \cr
& {\text{Calculate the average value}} \cr
& {f_{avg}} = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& {f_{avg}} = \frac{1}{{9 - 4}}\int_4^9 {\frac{1}{{\sqrt x }}} dx \cr
& {f_{avg}} = \frac{1}{5}\int_4^9 {{x^{ - 1/2}}} dx \cr
& {f_{avg}} = \frac{1}{5}\left[ {2\sqrt x } \right]_4^9 \cr
& {f_{avg}} = \frac{2}{5}\left[ {\sqrt 9 - \sqrt 4 } \right] \cr
& {f_{avg}} = \frac{2}{5} \cr
& \cr
& {\text{Let }}{f_{avg}} = f\left( x \right) \cr
& \frac{2}{5} = \frac{1}{{\sqrt x }} \cr
& \sqrt x = \frac{5}{2} \cr
& x = {\left( {\frac{5}{2}} \right)^2} \cr
& x = \frac{{25}}{4} \cr
& \cr
& {f_{avg}} = \frac{2}{5},{\text{ }}x = \frac{{25}}{4} \cr} $$
