Question

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

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

Answer 1

Answer: (B)

$f(x)=2x^3-9a{x}^2+12a^{2}x+1$
$\therefore$ $f^{\prime }(x)=6x^2-18ax+12a^2$
and $f^{\prime \prime }(x)=12x-18a$ For $\max /\min ,6x^2-18ax+12a^2=0$
$\Rightarrow$ $x^2-3ax+2a^2=0$
$\Rightarrow$ $(x-a)(x-2a)=0$
$\Rightarrow$ $x=a$ or $x=2a$
Now, $f^{\prime \prime }(a)=12a-18a=-6a<0$ and $f^{\prime \prime }(2a)=24a-18a=6a>0$
$\therefore$ $f(x)$ maximum at x = a and minimum at $x=2a$ .
$\Rightarrow$ $p=a$ and $q=2a$
Given that, $p^2=q$
$\Rightarrow$ $a^2=2a$
$\Rightarrow$ $a(a-2)=0$
$\therefore$ $a=2$