Answer
$$\eqalign{
& {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr
& y{\text{ - intercept: }}\left( {0,0} \right) \cr
& x{\text{ - intercepts: }}\left( {0,0} \right),\left( {\frac{{27}}{8},0} \right) \cr
& {\text{Relative maximum at }}\left( {1,1} \right) \cr
& {\text{No inflection points}} \cr
& {\text{No vertical asymptotes}} \cr
& {\text{No horizontal asymptotes}} \cr} $$
Work Step by Step
$$\eqalign{
& y = 3{x^{2/3}} - 2x \cr
& {\text{The domain of the function is }}\left( { - \infty ,\infty } \right) \cr
& \cr
& {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0 \cr
& y = 3{x^{2/3}} - 2x \cr
& y = 3{\left( 0 \right)^{2/3}} - 2\left( 0 \right) \cr
& y = 0 \cr
& y{\text{ - intercept }}\left( {0,0} \right) \cr
& \cr
& {\text{Find the }}x{\text{ intercept}}{\text{, let }}y = 0 \cr
& 0 = 3{x^{2/3}} - 2x \cr
& 0 = {x^{2/3}}\left( {3 - 2{x^{1/3}}} \right) \cr
& x = 0,{\text{ }}x = {\left( {\frac{3}{2}} \right)^3} = \frac{{27}}{8} \cr
& x{\text{ - intercepts }}\left( {0,0} \right),\left( {\frac{{27}}{8},0} \right) \cr
& \cr
& *{\text{Find the extrema}} \cr
& {\text{Differentiate}} \cr
& y' = \frac{d}{{dx}}\left[ {3{x^{2/3}} - 2x} \right] \cr
& y' = 2{x^{ - 1/3}} - 2 \cr
& {\text{Let }}y' = 0{\text{ to find critical points}} \cr
& - 2{x^{ - 1/3}} - 2 = 0 \cr
& {x^{ - 1/3}} + 1 = 0 \cr
& {x^{ - 1/3}}\left( {{x^{1/3}} + 1} \right) = 0 \cr
& {x^{1/3}} + 1 \to x = 1 \cr
& {\text{And is undefined at }}x = 0, \cr
& {\text{The critical points are: }}x = 0{\text{ and }}x = 1 \cr
& \cr
& *{\text{Find the second derivative}} \cr
& y'' = \frac{d}{{dx}}\left[ {2{x^{ - 1/3}} - 2} \right] \cr
& y'' = - \frac{2}{3}{x^{ - 4/3}} \cr
& {\text{Evaluate }}y''\left( x \right){\text{ at }}x = 1 \cr
& y''\left( 1 \right) = - \frac{2}{3}{\left( 1 \right)^{ - 4/3}} = - \frac{2}{3},{\text{ relative maximum at }}x = 1 \cr
& f\left( 1 \right) = 3{\left( 1 \right)^{2/3}} - 2\left( 1 \right) = 1 \to {\text{Relative maximum at }}\left( {1,1} \right) \cr
& \cr
& {\text{The derivative is not defined at }}x = 0 \cr
& {\text{Evaluate the first derivative test}} \cr
& f'\left( { - 0.5} \right) < 0,{\text{ }}f'\left( {0.5} \right) > 0 \cr
& {\text{Changes Negative to Positive}}{\text{, Relative minimum at }}x = 0 \cr
& f\left( 0 \right) = 0 \to {\text{ Relative minimum at }}\left( {0,0} \right) \cr
& \cr
& {\text{Let }}y''\left( x \right) = 0 \cr
& - \frac{2}{3}{x^{ - 4/3}} = 0 \cr
& {\text{The second derivative is undefined at }}x = 0,{\text{ then}} \cr
& y''\left( { - 1} \right) = - \frac{2}{3} < 0,{\text{ concave down}} \cr
& y''\left( 0 \right) = {\text{undefined}} \cr
& y''\left( {0.5} \right) \approx - 1.67 < 0,{\text{ concave down}} \cr
& x = 0{\text{ is not an inflection point}} \cr
& \cr
& {\text{*Calculate the asymptotes}} \cr
& {\text{No vertical asymptotes}}{\text{, the denominator is 1}}{\text{.}} \cr
& \mathop {\lim }\limits_{x \to \infty } \left( {3{x^{2/3}} - 2x} \right) = - \infty \cr
& \mathop {\lim }\limits_{x \to - \infty } \left( {3{x^{2/3}} - 2x} \right) = + \infty \cr
& {\text{No horizontal asymptotes}} \cr
& \cr
& {\text{Graph}} \cr} $$
