# How many turning points can a cubic function have?

Sep 6, 2014

Any polynomial of degree $n$ can have a minimum of zero turning points and a maximum of $n - 1$. However, this depends on the kind of turning point.

Sometimes, "turning point" is defined as "local maximum or minimum only". In this case:

• Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of $n - 1$.
• Polynomials of even degree have an odd number of turning points, with a minimum of 1 and a maximum of $n - 1$.

However, sometimes "turning point" can have its definition expanded to include "stationary points of inflexion". For an example of a stationary point of inflexion, look at the graph of $y = {x}^{3}$ - you'll note that at $x = 0$ the graph changes from convex to concave, and the derivative at $x = 0$ is also 0.

If we go by the second definition, we need to change our rules slightly and say that:

• Polynomials of degree 1 have no turning points.
• Polynomials of odd degree (except for $n = 1$) have a minimum of 1 turning point and a maximum of $n - 1$.
• Polynomials of even degree have a minimum of 1 turning point and a maximum of $n - 1$.

So, in part, it depends on the definition of "turning point", but in general most people will go by the first definition.