What is condition cubic polynomial

Question

We all know, $ax^2+bx+c \geqslant 0$ is true if $a > 0$ and $4ac-b^2 \geqslant 0.$ So, what is conditon of $$ax^3+bx^2+cx+d \geqslant 0, \quad a > 0 \quad ?$$

Answer 1

There is no such condition since $$\lim _{x\to \infty}f(x)= \infty$$ and $$\lim _{x\to -\infty}f(x)= -\infty$$

meaning $Range(f) = \mathbb{R}$ since it is continious.

Answer 2

$$ax^3+bx^2+cx+d \geqslant 0$$

This is a false statement for real coefficients of the equation. (I can't say for imaginary ones).

Letting $$ax^3+bx^2+cx+d=y$$ (y being the range).

The graph would go on from $-\infty$ to $+\infty$ as stated by Aqua.

The graph would look somewhat like this:—

Finding the coefficient of Cubic equation by using Cardano's Method can be helpful.