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.