The function $f(x)=2 x^3-3 x^2-12 x+4$ has:

A. two points of local maxima.
B. two points of local minima
C. one local maxima and one local minima.
D. no local maxima or local minima.

Choose the correct answer from the options given below:

C only

The correct answer is Option (1) → C only

$f(x)=2 x^3-3 x^2-12 x+4$

$f'(x)=6x^2-6x-12=0⇒x^2-x-2=0$

So $(x-2)(x+1)=0$

$x=-1,2$

$f''(x)=2x-1$

$f''(-1)<0$ → point of local maxima

$f''(2)>0$ → point of local minima