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

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Solution

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

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.

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