How many stationary points can a cubic function have?

1 Answer
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(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.