Question

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

• A. $2$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $3$

Answer 1

The correct option is A $2$

$f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,a>0f^{\prime }\left(x\right)=6x^2-18ax+12a^2=6(x^2-3ax+2a^2)=0\text{for extreme values}=6(x-a)(x-2a)=0\Rightarrow x=a,2af"\left(x\right)=12x-18a,f"\left(a\right)=-6a<0\text{Max at x = a = p}f"\left(2a\right)=6a>0\text{Min at x = 2a = q}\text{Given}{p}^2=q\Rightarrow a^2=2a\Rightarrow a=2$