Study of the cubic y = x cubed minus 12x

Text

Study the function $y=x^{3}-12x$: domain, symmetry, axis intercepts, monotonicity, relative extrema and inflection.

Solution

Domain $\mathbb{R}$; odd function. Intercepts: $x(x^{2}-12)=0\Rightarrow x=0,\ x=\pm2\sqrt3$. First derivative: $y'=3x^{2}-12=3(x-2)(x+2)=0\Rightarrow x=\pm2$. Increasing for $x<-2$ and $x>2$, decreasing on $(-2,2)$. Relative max $(-2,16)$, relative min $(2,-16)$. Second derivative: $y''=6x=0\Rightarrow x=0$: inflection at $(0,0)$. $\boxed{\text{max }(-2,16);\ \text{min }(2,-16);\ \text{inflection }(0,0)}$