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 1 Let
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
.