Answer
Graph
Work Step by Step
$$\eqalign{
& y = 3{x^4} + 4{x^3} \cr
& {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr
& {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr
& y = 3{\left( 0 \right)^4} + 4{\left( 0 \right)^3} \cr
& y = 0 \cr
& y{\text{ - intercept }}\left( {0,0} \right) \cr
& {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr
& 3{x^4} + 4{x^3} = 0 \cr
& {x^3}\left( {3x + 4} \right) = 0 \cr
& x = 0,{\text{ }}x = - \frac{4}{3} \cr
& x{\text{ - intercepts }}\left( {0,0} \right),\left( { - \frac{4}{3},0} \right) \cr
& \cr
& *{\text{Find the extrema}} \cr
& {\text{Differentiate}} \cr
& y' = \frac{d}{{dx}}\left[ {3{x^4} + 4{x^3}} \right] \cr
& y' = 12{x^3} + 12{x^2} \cr
& {\text{Let }}y' = 0{\text{ to find critical points}} \cr
& 12{x^3} + 12{x^2} = 0 \cr
& 12{x^2}\left( {x + 1} \right) = 0 \cr
& x = 0,{\text{ }}x = - 1 \cr
& *{\text{Find the second derivative}} \cr
& y'' = \frac{d}{{dx}}\left[ {12{x^3} + 12{x^2}} \right] \cr
& y'' = 36{x^2} + 24x \cr
& {\text{Evaluate }}y''\left( x \right){\text{ at }}x = 0{\text{ and }}x = - 1 \cr
& y''\left( { - 1} \right) = 36{\left( 0 \right)^2} + 24\left( 0 \right) = 0 \cr
& {\text{Using the first derivative test }} \cr
& y'\left( { - 0.5} \right) = + ,{\text{ }}y'\left( {0.5} \right) = + ,{\text{ The derivative does not change}} \cr
& {\text{no relative extrema at }}x = 0 \cr
& y''\left( { - 1} \right) = 36{\left( { - 1} \right)^2} + 24\left( { - 1} \right) = 12 > 0 \cr
& {\text{Relative minimum at }}x = - 1 \cr
& y\left( { - 1} \right) = 3{\left( { - 1} \right)^4} + 4{\left( { - 1} \right)^3} \to {\text{Relative minimum at }}\left( { - 1, - 1} \right) \cr
& \cr
& {\text{Let }}y''\left( x \right) = 0 \cr
& 36{x^2} + 24x = 0 \cr
& 12x\left( {3x + 2} \right) = 0 \cr
& x = 0,{\text{ }}x = - \frac{2}{3} \cr
& {\text{Inflection points at }}x = 0{\text{ and }}x = - \frac{2}{3} \cr
& \cr
& {\text{*Calculate the asymptotes}} \cr
& {\text{No vertical asymptotes, the denominator is 1}}{\text{.}} \cr
& \mathop {\lim }\limits_{x \to \infty } \left( {2 - x - {x^3}} \right) = - \infty \cr
& \mathop {\lim }\limits_{x \to - \infty } \left( {2 - x - {x^3}} \right) = + \infty \cr
& {\text{No horizontal asymptotes}} \cr
& \cr
& {\text{Graph}} \cr} $$
