Range of root of cubic equation.

Question

Suppose $a$ and $b$ are two positive real numbers such that the roots of cubic equation $x^3-ax+b=0$ are all real. Let $p$ be a root of this equation with minimal absolute value. What is the range of p. I tried to apply various sum and product of roots constraints but can't get it. Please help.

Answer 1

By Viète's theorem, the roots of such a polynomial have to be $\zeta_1,\zeta_2,\zeta_3=-(\zeta_1+\zeta_2)\in\mathbb{R}$ and

$$ \zeta_1\zeta_2(\zeta_1+\zeta_2)=b,\qquad \zeta_1\zeta_2-(\zeta_1+\zeta_2)^2=a\tag{1} $$ with the second condition representing an ellipse in the $(\zeta_1,\zeta_2)$ plane, with its minor and major axis on the $\zeta_1=\zeta_2$ and the $\zeta_1=-\zeta_2$ lines. It is not difficult to bound such ellipse into/around a rectangle and derive bounds for $|\zeta_1|,|\zeta_2|$ and $|\zeta_3|$.