If the function $f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1$, where $a0$, 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)]

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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 val…

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