Answer
$$\eqalign{
& {\text{critical points: }}\left( {0,0} \right),{\text{ }}\left( {1, - 1} \right),{\text{ }}\left( {1,1} \right) \cr
& {\text{inflection points: }}none \cr} $$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( x \right) = 2{x^2} - 3{x^{4/3}} \cr
& {\text{Differentiate }}f'\left( x \right) \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {2{x^2} - 3{x^{4/3}}} \right] \cr
& f'\left( x \right) = 4x - 4{x^{1/3}} \cr
& {\text{Calculate the critical points set the derivative equal to 0}} \cr
& 4x - 4{x^{1/3}} = 0 \cr
& 4{x^{1/3}}\left( {{x^{2/3}} - 1} \right) = 0 \cr
& 4{x^{1/3}} = 0,{\text{ }}{x^{2/3}} - 1 = 0 \cr
& x = 0,{\text{ }}x = \pm 1 \cr
& {\text{Calculate }}f\left( 0 \right),{\text{ }}f\left( { \pm 1} \right) \cr
& f\left( 0 \right) = 0,{\text{ }}f\left( { \pm 1} \right) = \pm 1 \cr
& {\text{Therefore, the critical points are:}} \cr
& \left( {0,0} \right),{\text{ }}\left( {1, - 1} \right),{\text{ }}\left( {1,1} \right) \cr
& \cr
& {\text{Calculate the second derivative}} \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {4x - 4{x^{1/3}}} \right] \cr
& f''\left( x \right) = 4 - \frac{4}{3}{x^{ - 2/3}} \cr
& f'\left( x \right) = - \frac{2}{{3{x^{4/3}}}} \cr
& \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& - \frac{2}{{3{x^{4/3}}}} = 0 \cr
& {\text{The second derivative is never 0, then there are no inflection}} \cr
& {\text{points}}{\text{.}} \cr} $$
