Originally Posted by
Soroban Hello, ignis!
I assume you mean a third-degree polynomial function.
The general cubic function is: .$\displaystyle f(x) \:=\:ax^3 + bx^2 + cx +d,\;\;a \ne 0 $
The first derivative is: .$\displaystyle f'(x) \:=\:3ax^2 + 2bx + c$
The second derivative is: .$\displaystyle f''(x) \:=\:6ax + 2b $
A point of inflection occurs where $\displaystyle f''(x) \,=\,0$
. . $\displaystyle 6ax + 2b \:=\:0 \quad\Rightarrow\quad x \:=\:-\dfrac{b}{3a}$
Therefore, there is an inflection point at: .$\displaystyle \left(-\dfrac{b}{3a},\:\dfrac{2b^3-9abc + 27a^2d}{27a^2}\right)$