Question:
Let $f(x)=2 x^{3}-3 x^{2}-12 x+5$ on $[-2,4]$. The relative maximum occurs at $x=$
(a) $-2$
(b) $-1$
(C) 2
(d) 4
Solution:
(c) 2
Given : $f(x)=2 x^{3}-3 x^{2}-12 x+5$
$\Rightarrow f^{\prime}(x)=6 x^{2}-6 x-12$
For a local maxima or a local minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow 6 x^{2}-6 x-12=0$
$\Rightarrow x^{2}-x-2=0$
$\Rightarrow(x-2)(x+1)=0$
$\Rightarrow x=2,-1$
Now,
$f^{\prime \prime}(x)=12 x-6$
$\Rightarrow f^{\prime \prime}(-1)=-12-6=-18<0$
So, $x=1$ is a local maxima.
Also,
$f^{\prime \prime}(2)=24-6=18>0$
So, $x=2$ is a local minima.