Solution for QID #95710: Any polynomial of degree ##n## can have a minimum of zero tu | StudyHelpMe

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.