Question

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,$ where a > 0, attains its maximum and minimum at p and q respectively such that $p^2=q$, then a equals

• A. 3
• B. 1
• C. 2
• D. 1/2

Answer 1

The correct option is C

2

We have, $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1\therefore f\left(x\right)=6x^2-18ax+12a^2=0\Rightarrow 6[x^2-3ax+2a^2]=0\Rightarrow x^2-3ax+2a^2=0\Rightarrow x^2-2ax-ax+2a^2=0\Rightarrow x(x-2a)-a(x-2a)=0\Rightarrow (x-a)(x-2a)=0\Rightarrow x=a,x=2a$
Now, $f^{\prime }\left(x\right)=12x-18a$
$\therefore f^{\prime }\left(a\right)=12a-18a=-6a<0\therefore f\left(x\right)$ will be maximum at x = a
i.e. p = a
Also, $f^{\prime }\left(2a\right)=24a-18a=6a\therefore f\left(x\right)$will be minimum at x = 2a
i.e.q = 2a
Given, $p^2=q$
$\Rightarrow a^2=2a\Rightarrow a=2.$
Hence (c) is the correct answer.