Theory of Equations

Given that all coefficients are positive, all real roots must be negative. Let these roots be $-\pi_{1},-\pi_{2},-\pi_{3}$ .

We know that the product $\pi_{1} \pi_{2} \pi_{3} =2$ and want to minimize the sum $\pi_{1}+\pi_{2}+\pi_{3}=\frac{a}{2}$ .

By using AM-GM inequality, $\frac{\pi_{1}+\pi_{2}+\pi_{3}}{3} \ge \sqrt [3] {\pi_{1} \pi_{2} \pi_{3}}$

$\huge a \ge 6.2^{\frac{1}{3}}$

Differentiating the given equation and equating it's discriminant = 0 . i.e. to get min value of 'a'

Then we get $a^2$ = 6b

Now for the equation $ax^3+bx^2+cx+d$

Discriminant is $b^2c^2-4ac^3-4ca^3-27a^2d^2+18abcd$ equating discriminant to 0 .

I.e. to get min value of a and substituting b= $a^2$ /6

We get discriminant as quadratic in $a^3$ and the roots of it are equal and they are 432,432.

Therefore we conclude that $a^3$ = 432.