Lagrange's Mean Value Theorem

From AoPSWiki

Lagrange's mean value theorem (often called "the mean value theorem," and abbrviated 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.

Statement

Let be a continuous function, differentiable on the open interval . Then there is 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.

Proof

We reduce the problem to the Rolle's theorem by using an 'auxillary function'.

Consider

note that

By Rolle's theorem, such that

i.e.

or

QED

See Also

• Rolle's theorem
• Derivative
• Fundamental Theorem of Calculus