Question

How many stationary points can a cubic function have?

Answer 1

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(x) = Ax^3+Bx^2+Cx+D` `{A,B,C,D}in RR`

Now consider `f'(x)`

`f'(x) = 3Ax^2+2Bx+C`

The stationary points of `f(x)` will be where `f'(x)=0`

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

Hence, `f(x)` will have at most 2 real stationary points.