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

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

Differentiate both sides with respect to $x$

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

Equate to zero, and then find the values 

$f\text{'}\left(x\right)=0$

It implies that

$\begin{array}{rcl}{x}^2-3ax+2a^2& =& 0\\ x^2-2ax-ax+2a^2& =& 0\\ x\left(x-2a\right)-a\left(x-2a\right)& =& 0\\ \left(x-a\right)\left(x-2a\right)& =& 0\end{array}$

Hence,

$x=a,x=2a$

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

By substituting $x=a$

$f\text{'}\text{'}\left(a\right)=4a>0$

therefore, the function has minimum value at $x=a=p$

therefore, $p=a, q=2a$

Given $p^2=q\Rightarrow a^2=2a\Rightarrow a=\boxed{\mathbf{2}}$