Question

State whether $-2$ is a point of local maxima or a point of local minima on the curve $y=2x^3+3x^2-12x+12.$

Answer 1

$y=2x^3+3x^2-12x+12$
$\Rightarrow \frac{dy}{dx}=6x^2+6x-12$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=12x+6$

If $\frac{d^{2}y}{d{x}^2}{|}_{x=a}<0,$ then $a$ is a point of local maxima and
if $\frac{d^{2}y}{d{x}^2}{|}_{x=a}>0,$ then $a$ is a point of local minima.

$\Rightarrow \frac{d^{2}y}{d{x}^2}{|}_{x=-2}=-18$
$\Rightarrow -2$ is a point of maxima.