# How many stationary points can a cubic function have?

Jun 10, 2018

A cubic polynomial with real coefficients can have at most 2 real stationary points

#### Explanation:

Here we will limit ourselves to a cubic polynomial with real coefficients.

Consider the general equation of a cubic polynomial:

$f \left(x\right) = A {x}^{3} + B {x}^{2} + C x + D$ $\left\{A , B , C , D\right\} \in \mathbb{R}$

Now consider $f ' \left(x\right)$

$f ' \left(x\right) = 3 A {x}^{2} + 2 B x + C$

The stationary points of $f \left(x\right)$ will be where $f ' \left(x\right) = 0$

$f ' \left(x\right)$ is a quadratic that will have 2 real or complex roots. The real roots may be co-incident.

Hence, $f \left(x\right)$ will have at most 2 real stationary points.