If the function $f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where $\mathrm{a} \gt 0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $\mathrm{p}^2=\mathrm{q}$, then $f(3)$ is equal to:

Question

Aicharya · Accepted Answer

$f^{\prime}(x)=6 x^2-18 a x+12 a^2$
$f^{\prime}(x)=6\left(x^2-3 a x+2 a^2\right)$
roots are $a, 2 a$
$\mathrm{p}^2=\mathrm{q} \Rightarrow \mathrm{a}^2=2 \mathrm{a}$
$a=2$
$f(x)=2 x^3-18 x^2+48 x+1$
$f(3)=37$

If the function $f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where $\mathrm{a} \gt 0$, attains…

Solution