JEE 2025 — Mathematics PYQ

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

$f^{\prime }(x)=6x^2-18ax+12a^2$

$f^{\prime }(x)=6(x^2-3ax+2a^2)$

$x^2-3ax+2a^2=0$

$x=a,,2a$

$f^{\prime \prime }(x)=6(2x-3a)$

f''(a)=6(2a-3a)=-6a<0

$\therefore \text{Local maxima at }x=a$

$\text{Local minima at }x=2a$

Given:

$p^2=q$

$a^2=2a$

$a(a-2)=0$

$a=0\text{ or }2$

\text{Since } a>0,\ a=2

$f(x)=2x^3-18x^2+48x+1$

$f(3)=37$