Question

The function $f\left(x\right)=2x^3-3x^2-36x+2$ has its maxima at

• A. $x=-2$ only
• B. $x=0$ only
• C. $x=3$ only
• D. Both $x=-2$ and $x=3$

Answer 1

The correct option is A $x=-2$ only

$f\left(x\right)=2x^3-3x^2-36x+2$
$f^{\prime }\left(x\right)=6x^2-6x-36$ .... (1)
For critical points:
$f^{\prime }\left(x\right)=0$
$\Rightarrow 6x^2-6x-36=0$
$\Rightarrow x^2-x-6=0$
$(x-3)(x+2)=0$
$x=3,-2$
$\Rightarrow f^{\prime \prime }\left(x\right)=12x-6$
At $x=3$
$f^{\prime \prime }\left(x\right)=30>0$
Hence $x=3$ point of minima.
at $x=-2$
$f^{\prime \prime }(-2)=-30<0$
$x=-2$ is point of maxima