Find the positive value of ' a ' for which the equality 2 α + β = 8 holds, where '…

Given, $a>0$

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

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

$f\text{'}\left(x\right)=6\left(x^2-3ax+2a^2\right)$

$f\text{'}\left(x\right)=6\left(x-2a\right)\left(x-a\right)$.

$x=a, x=2a$

$f\text{'}\text{'}\left(x\right)=6\left(2x-3a\right)$

$f\text{'}\text{'}\left(a\right)=6\left(2a-3a\right)<0$

$f\text{'}\text{'}\left(2a\right)=6\left(4a-3a\right)>0$

Thus, $\alpha =a, \beta =2a$

$\Rightarrow 2\alpha +\beta =8$

$\Rightarrow 2a+2a=8$

$\Rightarrow 4a=8$

$\Rightarrow a=2$