Question

Suppose $x_1$ & $x_2$ are the point of maximum and the point of minimum respectively of the function $f\left(x\right)=22x^3-9a{x}^2+12a^{2}x+1$ respectively, then for the equality $x_1^2=x_2$ to be true the value of ${}^{\prime }{a}^{\prime }$ must be

• A. $0$
• B. $a=\frac{3+\sqrt{5}}{(3-\sqrt{5})}$
• C. $a=\frac{3+\sqrt{5}}{{(3-\sqrt{5})}^2}$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=22x^3-9a{x}^2+12a^{2}x+1f^{\prime }\left(x\right)=66x^2-18ax+24a^2$
Discernment for this is
$D={\left(18a\right)}^2-4\left(24a^2\right)\left(66\right)=324a^2-6336a^2=-6012a^2<0$
So, roots are complex, this implies that $f^{\prime }\left(x\right)$ is always positive and $f\left(x\right)$ is monotonically increasing and no maxima and minima exits.