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