# The Mean Value Theorem and its Applications.

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The mean value theorem belongs to one of the most used and important theorems in analysis. Its statement is very intuitive and moreover surprisingly easy to prove. In this story I quickly want to describe the main idea around this theorem and to consider some of its direct applications.

## The mean value theorem:

This is already the non-formal version of the mean value theorem. In the following we will focus on giving a precise formulation and how to prove it.

Next we are making some observations that relate to the derivative of `f`

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Next, let us try to apply the previous result to find proof for the initially stated problem.

## Some applications:

When you deal with differential equations, sooner or later you will encounter this:

In conjunction with interpolation polynomials the following is often taken into account:

Beside the case of real-valued function in one variable, the mean value theorem can be generalized tremendously.

Thanks for reading!