If your average speed was 60, at some instant you were doing exactly 60
Drive smoothly from one town to another and, somewhere along the way, your speedometer must have shown precisely your average speed for the trip. That everyday certainty is the mean value theorem, one of calculus's workhorses, with roots stretching from a fifteenth-century Kerala astronomer to Augustin-Louis Cauchy.
Stated carefully: take a function that is continuous on a closed interval and differentiable inside it. Then at some interior point, its instantaneous rate of change equals its average rate of change across the whole interval. Picture the graph. Draw a straight chord between the two endpoints; somewhere between them, the curve's tangent runs exactly parallel to that chord.
Its special case is Rolle's theorem: if the function starts and ends at the same height, then at some point in between its slope is zero, a flat spot at a peak or trough. The general result follows by a neat trick. Subtract from the function a straight line with the chord's slope, which levels the endpoints, and Rolle's theorem hands you the point you need. The conditions matter, though they can be relaxed somewhat. The cube root function still qualifies even though its slope shoots to infinity at the origin, and only continuity, not differentiability, is required at the endpoints.
The history is spread across centuries. Parameshvara, who lived from 1380 to 1460 and belonged to the Kerala school of astronomy and mathematics, described a special case in connection with interpolating the sine function. Michel Rolle proved a limited version in 1691, and only for polynomials, without using calculus at all. Cauchy finally gave the general statement, with a proof, in 1823.
Cauchy also gave an extended version that compares two functions at once, so that the ratio of their rates of change at some point matches the ratio of their overall changes. Geometrically this concerns a curve traced out by a pair of functions, though the promised tangent can fail to exist where the curve stalls. The theorem's real value is as a tool, underpinning proofs of many general facts about differentiable functions.
Source: Mean value theorem
