How many stationary points can a cubic function have?

1 Answer
Jun 10, 2018

Answer:

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.