Can a polynomial have two distinct real zeros and no local extreme? If yes, please explain. Can a polynomial have local extremes without having any real zeros? If yes, please explain.

Question

Can a polynomial have two distinct real zeros and no local extreme? If yes, please explain.

Can a polynomial have local extremes without having any real zeros? If yes, please explain.

Answer 1

In the first case, No. if the polynomial has two distinct real zeros then it is at least a quadratic and will have a minimum of 1 local max or min.

In the second case, yes. The polynomial must be even power in it's degree so that it doesn't cross through the x axis on either end . Think of a simple case y=x^2+1. This quadratic has no real roots and has a local minimum.