Question

If $x=p$ and $x=q$ are the points of maxima and minima of $f\left(x\right)=x^3-12a^{2}x+36a{x}^2-4$, $(a>0)$ then

• A. $p>q$
• B. $p<q$
• C. $p=q$
• D. $2p>q$

Answer 1

The correct option is B $p<q$

$f\left(x\right)=x^3-12a{x}^2+36a^{2}x-4$, $(a>0)$
$f^{\prime }\left(x\right)=3x^2-24ax+36a^2$
$f"\left(x\right)=6x-24a$
$f"\left(x\right)=6(x-4a)$
$f"\left(p\right)<0$ $\Rightarrow$ $p-4a<0$ $\Rightarrow$ $p<4a$
$f"\left(q\right)<0$ $\Rightarrow$ $q-4a<0$ $\Rightarrow$ $q>4a$
$p<4a<q$ $\therefore$ $p<q$
$\therefore$ (2) is correct