What is the ratio of the relative minimum to the relative maximum value of a cubic?

Question

What is the ratio of the relative minimum to the relative maximum value of the general cubic polynomial $ax^3+bx^2+cx+d$? Consider only cubic functions with $3$ real, unique roots.

Answer 1

Set the derivative, $3ax^2+2bx+c=0$ to find relative extrema. Solving for $x$, we find that $$x_{extrema}=\frac{-b\pm\sqrt{{{b}^{2}}-3ac}}{3a}$$ Let $f(x)=ax^3+bx^2+cx+d$. Now, we just need to simplify $$\frac{f\left(x_{extrema}^+\right)}{f\left(x_{extrema}^-\right)}$$ which comes out to $$-\frac{-2 {{b}^{3}}+9 a b c+\sqrt{{{b}^{2}}-3 a c} \left( 6 a c-2 {{b}^{2}}\right) -27 {{a}^{2}} d}{27 {{a}^{2}} d+\sqrt{{{b}^{2}}-3 a c} \left( 6 a c-2 {{b}^{2}}\right) -9 a b c+2 {{b}^{3}}}$$ It is worthwhile to note that the constraint ${27 {{a}^{2}} d+\sqrt{{{b}^{2}}-3 a c} \left( 6 a c-2 {{b}^{2}}\right) -9 a b c+2 {{b}^{3}}}\ne0$ is satisfied by the restriction that the cubic must have $3$ distinct real roots. As well, the constraint $3a\ne0$ is also satisfied by the fact that the polynomial must be a cubic.

The above expression can be simplified with a number of substitutions. Let $\alpha=2b^3-9abc+27a^2d$ and $\beta=\sqrt{b^2-3ac}\left(6ac-2b^2\right)$.

Then, the ratio of the relative minimum to relative maximum of the cubic is simply $$-\frac{\beta-\alpha}{\beta+\alpha}$$