For $a0$, if the function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ attains its maximum value at $p$ and minimum value at $q$ such that $p^2=q$, then $a=$

Question

For $a>0$, if the function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ attains its maximum value at $p$ and minimum value at $q$ such that $p^2=q$, then $a=$

Aicharya · Accepted Answer

Given, $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ has maximum value at $P$ and minimum value at $q$. So, $f^{\prime}(p)=0$ and $f^{\prime}(q)=0$ Now, $f^{\prime}(x)=6 x^2-18 a x+12 a^2$ has roots $p$ and $q$ $ \begin{aligned} & 	herefore & p+q & =\frac{18 a}{6}=3 a \ & 	ext { and } & p q & =\frac{12 a^2}{6}=2 a^2 \end{aligned} $ on solving both equations, we get $ \begin{aligned} & p=a 	ext { and } q=2 a \ & 	ext { Given, } \ & p^2=q \ & 	herefore \quad a^2=2 a \ & \Rightarrow \quad a^2-2 a=0 \ & \Rightarrow \quad a(a-2)=0 \ & \Rightarrow \quad a=2 \quad(\because a 
eq 0) \ & \end{aligned} $

For $a>0$, if the function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ attains its maximum value at $p$ and minimum…

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