Answer
$$\eqalign{
& {\text{critical points: }}\left( {0,0} \right),{\text{ }}\left( { - 1,1} \right) \cr
& {\text{inflection points: }}none \cr} $$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( x \right) = 2x + 3{x^{2/3}} \cr
& {\text{Differentiate }}f'\left( x \right) \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {2x + 3{x^{2/3}}} \right] \cr
& f'\left( x \right) = 2 + 3\left( {\frac{2}{3}} \right){x^{2/3 - 1}} \cr
& f'\left( x \right) = 2 + \frac{2}{{{x^{1/3}}}} \cr
& {\text{Calculate the critical points set the derivative equal to 0}} \cr
& 2 + \frac{2}{{{x^{1/3}}}} = 0 \cr
& \frac{2}{{{x^{1/3}}}} = - 2 \cr
& x = - 1 \cr
& {\text{The derivative is not defined when }}{x^{1/3}} = 0,{\text{ then}} \cr
& x = 0 \cr
& {\text{Calculate }}f\left( 0 \right),{\text{ }}f\left( { - 1} \right) \cr
& f\left( 0 \right) = 0,{\text{ }}f\left( { - 1} \right) = 1 \cr
& {\text{Therefore, the critical points are:}} \cr
& \left( {0,0} \right),{\text{ }}\left( { - 1,1} \right) \cr
& \cr
& {\text{Calculate the second derivative}} \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {2 + \frac{2}{{{x^{1/3}}}}} \right] \cr
& f''\left( x \right) = 0 - \frac{2}{3}{x^{ - 4/3}} \cr
& f''\left( x \right) = - \frac{2}{{3{x^{4/3}}}} \cr
& \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& - \frac{2}{{3{x^{4/3}}}} = 0 \cr
& {\text{The second derivative is never 0, then there are no inflection}} \cr
& {\text{points}}{\text{.}} \cr} $$
