Example 6. Study the maximum and minimum values of the function $y=5-36 x+3 x^{2}+4 x^{3}$ on $(-2,+\infty)$.
Solve $\because y=x^{3}\left(\frac{5}{x^{3}}-\frac{3 b}{x^{2}}+\frac{3}{x}+4\right)$, when $x \rightarrow+\infty$, $x^{3} \rightarrow+\infty, \frac{5}{x^{3}}-\frac{36}{x^{2}}+\frac{3}{x}+4 \rightarrow 4$, $\therefore \mathrm{y}(+\infty)=+\infty$ (1), so $\mathrm{y}$ has no maximum value on $(-2,+\infty)$. $\mathrm{y}(-2+\dot{0})=\mathrm{y}(-2)=57$, and $y^{\prime}=-36+6 x+12 x^{2}=5\left(2 x^{2}+z-6\right)$ $=6(x+2)(2 x-3)$, in $(-2,+\infty)$ there is only one stationary point $x=\frac{3}{2}$, so $y$ has only a minimum value $y \left(\frac{3}{2}\right)=-\frac{115}{4}$ (Figure 8).