Consider the function [math] on the interval [math]. The Intermediate Value Theorem guarantees that for certain values of [math] there is a number [math] such that [math]. In the case of the function above, what, exactly, does the intermediate value theorem say? To answer, fill in the following mathematical statements, giving an interval with non-zero length in each case.

For every [math] in the interval [math] ,
there is a [math] in the interval [math]
such that [math].