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 equals:

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

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

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

$\therefore f^{\text{'}}\left(x\right)=0\Rightarrow x=a,2a$

$\therefore f^{\text{'}}\left(a\right)<0$ and $f^{\text{'}\text{'}}\left(2a\right)>0$

$\therefore f\left(a\right)$ Has max. at x = a and min. at x = 2a

$\therefore p=a,q=2a$

$\therefore a^2=2a\Rightarrow a=2\left(\because a>0\right)$