Lagrange's Mean Value Theorem
Lagrange's mean value theorem (often called "the mean value theorem," and abbreviated MVT or LMVT) is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using LMVT.
Let be a continuous function, differentiable on the open interval . Then there exists some such that .
Informally, this says that a differentiable function must at some point grow with instantaneous velocity equal to its average velocity over an interval.
We reduce the problem to Rolle's theorem by using an auxiliary function.
Consider Note that By Rolle's theorem, there exists in such that or $ which simplifies to as desired.