Question

How many turning points can a cubic function have?

Answer 1

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.