How many turning points can a cubic function have?

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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.

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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.

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