Question

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

• A. $1$
• B. $2$
• C. $1/2$
• D. $3$

Answer 1

The correct option is B $2$

Given, $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$

Thus $f^{\prime }\left(x\right)=6x^2-18ax+12a^2$

and $f^{\prime \prime }\left(x\right)=12x-18a$

For max/ min, $6x^2-18ax+12a^2=0$

$\Rightarrow x^2-3ax+2a^2=0$

$\Rightarrow (x-a)(x-2a)=0$

$\Rightarrow x=a$ or $x=2a$

Now, $f^{\prime \prime }\left(a\right)=12a-18a=-6a<0$

and $f^{\prime \prime }\left(2a\right)=24a-18a=6a>0$

Therefore, $f\left(x\right)$ is maximum at $x=a$ and minimum at $x=2a$.

$\Rightarrow p=a$ and $q=2a$

Given that, $p^2=q\Rightarrow a^2=2a$

$\Rightarrow a=2(a>0)$.