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.

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!