Do all polynomials have at least one minimum?

Answer 1

Polynomials of an even degree will always have either a minimum point, or a maximum point, or both.

Polynomials of an odd degree may or may not have minima or maxima. If, for example, a polynomial function is simply a transformation of xn, there will be no turning points. For example:

f(x) = x5 + 5x4 + 10x3 + 10x2 + 5x + 1 = (x+1)5

f'(x) = 5(x+1)4

There is only one solution for f'(x) = 0, which is of course x = -1. Since the range of f(x) includes all the real numbers, it follows that this solution represents a point of inflection, and not a turning point.

If a polynomial of odd degree does have any turning points, it will have at least one minimum point. It cannot have maximum points only.

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Polynomials of an odd degree cannot have a global maximum or minimum because if the leading coefficient is positive, it goes asymptotically from minus infinity to plus infinity and the other way around if the leading coefficient is negative.