Question

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$, $a>0$ attains its maximum and minimum value at $p$ and $q$ respectively. such that $p^2=q$, then ${}^{\prime }{a}^{\prime }$ equals

• A. $1$
• B. $2$
• C. $\frac{1}{2}$
• D. $3$

Answer 1

The correct option is B $2$

For maximum and minima $f^{\prime }\left(x\right)=0$
$\Rightarrow$ $6x^2-18ax+12a^2=0$ and $f^{\prime \prime }\left(x\right)=12x-18a$
$f^{\prime }\left(x\right)=0$
$\Rightarrow$ $x=a,2a$ and $f^{\prime \prime }\left(a\right)<0$ and $f^{\prime \prime }\left(2a\right)>0$
Now $p=a$ and $p=2a$
and $p^2=q$
$\Rightarrow$ $a^2=2a$
$\Rightarrow$ $a=0,a=2$
Ans: B