Which theorem indicates that a polynomial function must have a zero between two points if it changes signs at those points?

The Intermediate Value Theorem is a fundamental concept in calculus that applies to continuous functions. It states that if a function is continuous on a closed interval [a, b] and takes different signs at the endpoints (meaning f(a) and f(b) are of opposite signs), then there exists at least one point c within the interval (a, b) where the function f(c) is equal to zero.

In the context of polynomial functions, which are continuous everywhere, this theorem assures us that when a polynomial changes signs between two points, it must cross the x-axis, indicating the presence of a zero in that interval. This is crucial when analyzing the behavior of polynomials and determining their roots, as it not only indicates the existence of a root but also helps in narrowing down where to look for it.

The other theorems listed don't address this concept directly: the Fundamental Theorem of Algebra primarily concerns the number of roots a polynomial can have, the Mean Value Theorem relates to the average rate of change of a function, and the Extreme Value Theorem deals with the existence of maximum and minimum values of a function on a closed interval, none of which implies the existence of a zero based on sign changes.