Problem: Let $x^3 + ax^2 + bx + c$ be a polynomial with real nonnegative roots. Prove that b>= 3[(c^(2/3)]
I turned the equation into b/3 >= cuberoot(c^2) so that it's clearly a use of AM GM, but I'm not sure where to go from here.
Any help or a small hint would be appreciated, thanks :)
Your idea of using the inequality on AM and GM is correct.
Let $\alpha, \beta, \gamma$ be the real nonnegative roots of the given polynomial,
thus, $x^3+ax^2+bx+c = (x-\alpha)(x-\beta)(x-\gamma)$. By Vieta's formula,
$\alpha\beta+\beta\gamma+\gamma\alpha = b$ and $\alpha\beta\gamma=c$.
Can you do the rest?