Question

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$ has a local maximum at $x=x_1$ and a local minimum at $x=x_2$ such that $x_2=x_1^2$ then the value of ${}^{\prime }{a}^{\prime }$ equals.

• A. $0$
• B. $\frac{1}{2}$
• C. $2$
• D. Either (a) or (c)

Answer 1

The correct option is C $2$

Given $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$
$\Rightarrow f^{\prime }\left(x\right)=0$ for maximum or minimum
$\Rightarrow 6x^2-18ax+12a^2=0$
$\Rightarrow x=a,2a$(by observation)
Also $x_2=x_1^2$ (given)
$\Rightarrow a$ can not be $<0$
Now if $a>0$ then local minimum occurs at $x=2a$ and local maximum at $x=a$ so $x_1=a$ and $x_2=2a$. Using $x_2=x_1^2$
$\Rightarrow 2a=a^2\Rightarrow a^2-2a=0$
$\Rightarrow a=0,a=2\because a\ne 0$
$\Rightarrow a=2$ is the required value of $a$
Hence choice (c) is correct answer.