Answer
$x = 0,{\text{ }}x = 4,{\text{ }}x = \frac{4}{3}$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^{1/3}}{\left( {4 - x} \right)^{2/3}} \cr
& {\text{Differentiate with respect to }}x \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^{1/3}}{{\left( {4 - x} \right)}^{2/3}}} \right] \cr
& {\text{Using the product rule for derivatives}} \cr
& f'\left( x \right) = {x^{1/3}}\frac{d}{{dx}}\left[ {{{\left( {4 - x} \right)}^{2/3}}} \right] + {\left( {4 - x} \right)^{2/3}}\frac{d}{{dx}}\left[ {{x^{1/3}}} \right] \cr
& {\text{Computing derivatives and simplify}} \cr
& f'\left( x \right) = {x^{1/3}}\left( {\frac{2}{3}} \right){\left( {4 - x} \right)^{ - 1/3}}\left( { - 1} \right) + {\left( {4 - x} \right)^{2/3}}\left( {\frac{1}{3}{x^{ - 2/3}}} \right) \cr
& f'\left( x \right) = - \frac{2}{3}{x^{1/3}}{\left( {4 - x} \right)^{ - 1/3}} + \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{2/3}} \cr
& {\text{Factoring}} \cr
& f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {2x - \left( {4 - x} \right)} \right) \cr
& f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {2x - 4 + x} \right) \cr
& f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {3x - 4} \right) \cr
& f'\left( x \right) = - \frac{{3x - 4}}{{3{x^{2/3}}{{\left( {4 - x} \right)}^{1/3}}}} \cr
& {\text{The derivative does not exist when the denominator is 0}} \cr
& 3{x^{2/3}} = 0 \to x = 0 \cr
& or \cr
& {\left( {4 - x} \right)^{1/3}} = 0 \to x = 4 \cr
& {\text{Find the values of }}x{\text{ where }}f'\left( x \right) = 0 \cr
& - \frac{{3x - 4}}{{3{x^{2/3}}{{\left( {4 - x} \right)}^{1/3}}}} = 0 \cr
& 3x - 4 = 0 \cr
& x = \frac{4}{3} \cr
& \cr
& {\text{Applying the definition of critical numbers:}} \cr
& {\text{A critical number of a function is a number }}c{\text{ in the domain }} \cr
& {\text{of such that either }}f'\left( c \right) = 0{\text{ or }}f'\left( c \right){\text{ does not exist}}{\text{, so the}} \cr
& {\text{critical numbers are:}} \cr
& x = 0,{\text{ }}x = 4,{\text{ }}x = \frac{4}{3} \cr} $$