Question

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

Answer 1

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

$f^{\prime }\left(x\right)=6x^2-13mx+12m^2$

$\therefore f^{\prime }\left(x\right)=0;6x^2-18mx+12m^2=0$

$\therefore x^2-3mx+2m^2=0$

$\therefore (x-2m)(x-m)=0$

$\therefore p=mq=2m$

$\therefore p^2=q$

$\therefore m^2=2m$

$\therefore m^2-2m=0$

$\therefore m(m-2)=0$

$\therefore m>0;m=2$