JEE Mains · Maths · STD 12 - 6. Application of derivatives
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:
- A 55
- B 10
- C 23
- D 37
Answer & Solution
Correct Answer
(D) 37
Step-by-step Solution
Detailed explanation
\(f^{\prime}(x)=6 x^2-18 a x+12 a^2\) \(f^{\prime}(x)=6\left(x^2-3 a x+2 a^2\right)\) roots are \(a, 2 a\) \(\mathrm{p}^2=\mathrm{q} \Rightarrow \mathrm{a}^2=2 \mathrm{a}\) \(a=2\) \(f(x)=2 x^3-18 x^2+48 x+1\) \(f(3)=37\)
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