Question

Suppose $x_1$ and $x_2$ are the points of maximum and the point of minimum respectively of the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$ respectively, $(a>0)$ then for the equality ${x_1}^2=x_2$ to be true the value of $a$ must be

Answer 1

Local maxima and minima:

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

$f\text{'}\left(x\right)=0$

$\Rightarrow 6x^2–18ax+12a^2=0$

$\Rightarrow x^2–3ax+2a^2=0$

$\Rightarrow x^2–2ax-ax+2a^2=0$

$\Rightarrow x(x–2a)-a(x-2a)=0$

$\Rightarrow (x–a)=0or(x-2a)=0$

$\Rightarrow (x–a)=0or(x-2a)=0$

$\Rightarrow x=aor2a$

Let $x_1=2a,x_2=a$

$f\text{'}\text{'}\left(x\right)=12x–18a$

$f\text{'}\text{'}\left(2a\right)=24a-18a=6a>0\left(minima\right)$

$f\text{'}\text{'}\left(a\right)=12a–18a=-6a<0\left(maxima\right)$

Given $x_1$ and $x_2$ are the point of maximum and the point of minimum.

$x_2=2a$ is a minima and $x_1=a$ is a maxima.

${x_1}^2=x_2$

$\Rightarrow a^2=2a$

$\Rightarrow a=2$

Hence, the value of $ais2$