What conditions on the coefficients of $p(x) = ax^3 + bx^2 + cx + d$ guarantee that $p(x)\geq 0$ for some $x$?

Question

I am working on a research level problem and I am stuck up with a third degree polynomial inequality. So I would like to know:

Given the polynomial $p(x) = ax^3 + bx^2 + cx + d$, which conditions on the coeficients $a, b, c, d$ should we impose in order to have: $$p(x) \geq 0$$ for some $x$?

Answer 1

Note that if $a$ is nonzero, then the cubic tends to $-\infty$ on one side and $\infty$ on the other and the condition is satisfied. Otherwise, we have a quadratic; if $b$ is positive then we tend to $\infty$ and the condition is satisfied. If $b$ is negative, we require at least one root to satisfy the condition, i.e. the discriminant is nonnegative. If $b=0$, then if $c$ is nonzero we clearly satisfy the condition with a linear function. Finally, if $a = b = c = 0$ then $d \ge 0$ trivially works.

Answer 2

This is always true. If $a>0$ then as $x$ tends to positive infinity, we’re good. Otherwise, $a<0$ (as it’s a third degree polynomial) so as $x$ tends to negative we’re good.