If the function $f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1$, where a $0$, 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: [JEE Main 2025, 2 Apr (Shift 1)]

Question

If the function $f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1$, where a $>0$, 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: [JEE Main 2025, 2 Apr (Shift 1)]

YESPYQ · Accepted Answer

Correct answer: d) $37$. The first derivative of $f(x)$ is: ${f}^{′}(x)=6{x}^{2}−18ax+12{a}^{2}$ Factor the derivative: ${f}^{′}(x)=6({x}^{2}−3ax+2{a}^{2})=6(x−a)(x−2a)$ Setting ${f}^{′}(x)=0$ gives critical points $x=a$ and $x=2a$. Analyze the sign of ${f}^{′}(x)$: For $x 0$ (function increasing). For $a For \(x>2a$, ${f}^{′}(x)>0$ (function increasing). Thus, $x=a$ is a local maximum ( $p=a$ ) and $x=2a$ is a local minimum ( $q=2a$ ). Given ${p}^{2}=q$, substitute $p=a$ and $q=2a$: ${a}^{2}=2a$ Since $a>0$, divide both sides by $a$: $a=2$ Substitute $a=2$ into $f(x)$: $f(x)=2{x}^{3}−9(2){x}^{2}+12(2{)}^{2}x+1=2{x}^{3}−18{x}^{2}+48x+1$ Now, evaluate $f(3)$: $f(3)=2(3{)}^{3}−18(3{)}^{2}+48(3)+1$ Thus, $f(3)=37$

If the function \(f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1\), where a \(>0\), attains its local maximum and local minimum va…

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