The following theorem, which has very important consequences in differential calculus:
Let f be a function that is continuous on [a, b] and differentiable in (a, b). Then there is a number c with a < c < b such that
The result stated in the theorem can be expressed as a statement about the graph of f: if A, with coordinates (a, f(a), and B, with coordinates (b, f(b), are the points on the graph corresponding to the end-points of the interval, there must be a point C on the graph between A and B at which the tangent is parallel to the chord AB.
The theorem is normally deduced from Rolle's Theorem, which is in fact the special case of the Mean Value Theorem in which f(a)=f(b). A rigorous proof of either theorem relies on the non-elementary result that a continuous function on a closed interval attains its bounds. The Mean Value Theorem has immediate corollaries, such as the following. With the appropriate conditions on f,