According to the Intermediate Value Theorem, what condition must be met to ensure there is at least one zero of f between a and b?

The Intermediate Value Theorem states that if a function is continuous on a closed interval ([a, b]) and takes on different signs at the endpoints of that interval (i.e., (f(a)) and (f(b)) are of opposite signs), then there must be at least one value (c) within the interval ((a, b)) such that (f(c) = 0). This arises because a continuous function must traverse from one value to another without "jumping" over any values in between, which implies it will cross the x-axis at least once if it changes signs.

When (f(a)) is positive and (f(b)) is negative (or vice versa), it guarantees that there is a transition through zero. This condition is essential in confirming the presence of a root within the interval. In cases where (f(a)) and (f(b)) are both positive or both negative, there could be no crossing through the x-axis, thereby not ensuring any zero exists within that interval. Thus, the requirement that (f(a)) and (f(b)) be of opposite signs is crucial for applying the Intermediate Value Theorem effectively.