If the function \(f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1[a>0]\) attains its maximum and minimum at \(p\) and \(q\) respectively such that \(p^{2}=q\), then \(a\) is equal to

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If the function \(f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1[a>0]\) attains its maximum and minimum at \(p\) and \(q\) respectively such that \(p^{2}=q\), then \(a\) is equal to

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kritika · Answer 1 · 2021-12-26T08:16:46+0000

Ans : (A)
Hint : \(f^{\prime}(x)=6 x^{2}-18 a x+12 a^{2} \Rightarrow f^{\prime \prime}(x)=12 x-18 a \Rightarrow f^{\prime}(x)=0 \Rightarrow x=a, 2 a\)
\(f^{\prime \prime}(a)<0 ; p=a\) (maximum)
\(\mathrm{f}^{\prime \prime}(2 \mathrm{a})>0 ; \mathrm{q}=2 \mathrm{a}\) (minimum)
\(a^{2}=2 a ; a(a-2)=0, \quad a=2\)

If the function \(f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1[a>0]\) attains its maximum and minimum at \(p\) and \(q\) respectively such that \(p^{2}=q\), then \(a\) is equal to

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