In the first volume of Essential Geometry (1635), cavalieri, an Italian, gave an interesting lemma to deal with the tangents of plane and solid figures, in which Lemma 3 also described the same fact based on the geometric point of view: there must be tangents on the curve segments parallel to the curve chords. This is a geometric differential mean value theorem called cavalieri theorem. This theorem is the expression of Lagrange mean value theorem in geometry.
1797, Lagrange, a French mathematician, gave Lagrange's theorem for the first time in his book On Analytic Functions. The original form of his theorem is: "If the function is continuous between and, if there is a minimum and a maximum between and, it must take a value between and." Lagrange gave the initial proof, but the proof was not strict. The terms he offered are stronger than the current ones. He asked the function to have a continuous derivative in the closed interval, and the continuity he used was intuitive, not abstract.
/kloc-At the beginning of the 9th century, Cauchy gave a strict proof of Lagrange's mean value theorem in the strict movement of calculus. In introduction to infinitesimal calculation, Cauchy proved that "if the derivative is continuous in a closed interval, there must be a point that makes. Cauchy also extended Lagrange mean value theorem to Cauchy mean value theorem in the process of differential calculation.
The modern form of Lagrange's mean value theorem was given by French mathematician O.Bonnet, who used Rolle's theorem instead of the continuity of derivatives to prove Lagrange's mean value theorem again.