Question related to cubic equation.

Question

We have to find the minmum vale of ab if roots of the equation $x^3 -ax^2 +bx-2=0$ are positive.

I know the concept to solve for quadratic , but confused in this .

Find m with which quadratic equation has 2 positive answers

Answer 1

Let $p$, $q$, $r$ be the roots of the equation.

$$\begin{cases} p+q+r &=& a \\ pq+qr+pr &=& b \\ pqr &=& 2 \end{cases}$$

Thus:

$$ab = (p+q+r)(pq+qr+pr) \ge 9\sqrt[3]{pqr}\sqrt[3]{pqqrpr} = 9pqr = 18$$

With equality when $p=q=r=\sqrt[3]2$.