Maximum and minimum of a polynomial

Text

Find the values of $x$ at which $y=x^{3}-12x$ has a relative maximum or minimum, and the corresponding values of $y$.

Solution

$y'=3x^{2}-12=0\Rightarrow x=\pm2$. With $y''=6x$:

• at $x=-2$, $y''<0$: relative maximum, $y(-2)=16$;
• at $x=2$, $y''>0$: relative minimum, $y(2)=-16$. $\boxed{\text{max }(-2,\,16);\qquad \text{min }(2,\,-16)}$