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 to

Answer 1

The given equation is:

$f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$

To find the extremum points we differentiate and equate it to zero

$\Rightarrow f^{\prime }\left(x\right)=6x^2-18ax+12a^2$

$f^{\prime }\left(x\right)=0$

$\Rightarrow 6x^2-18ax+12a^2=0$

$\Rightarrow (x-2a)(x-a)=0$

$\Rightarrow x=2a,a$

Now to find whether at the critical points we find a maxima or minima we use the second derivative test.

$\Rightarrow f^{\prime \prime }\left(x\right)=12x-18a$

$\Rightarrow f^{\prime \prime }\left(2a\right)=24a-18a=6a$

$\Rightarrow f^{\prime \prime }\left(2a\right)>0$

$\Rightarrow f^{\prime \prime }\left(a\right)=12a-18a=-6a$

$\Rightarrow f^{\prime \prime }\left(a\right)<0$

Hence we get a minima at x =2a and and a maxima at x =a. Hence,

$\Rightarrow p=a$

$\Rightarrow q=2a$

Using the given relation we get

$p^2-q=0$

$\Rightarrow a^2-2a=0$

$\Rightarrow a(a-2)=0$

$\Rightarrow a\ne 0$ as $a>0$

$\Rightarrow a=2$ .....Answer