Christine Riedtmann asked me about the proof of the general mean value theorem mentioned in the very first entry of this blog. Let me recall the statement. In what follows is an interval and are (pairwise distinct) points on .

Theorem 1Let be times differentiable functions on . Then

with some .

Here is the Vandermonde determinant.

The proof is based on the more special Schwarz mean value theorem, also mentioned in that entry. It is more convenient for us to state the Schwarz theorem as follows.

Theorem 2 (Schwarz)Let be an times differentiable function on . Denote by the Lagrange interpolation polynomial for at the points (the polynomial of degree at most~ taking value at for .) Then for some we have .

In this form the Schwarz theorem is immediate: the “auxiliary function” vanishes at , and by the Rolle theorem its -th derivative vanishes somewhere on .

To prove Theorem 1 we consider the interpolation polynomials of at . Let be independent indeterminates. Elementary transformations of determinants show that

Applying to the both sides the differential operator

we obtain

Now the Schwarz theorem implies that with a suitably chosen .