The following proposition is often taught without proof. A nice proof is writtenusing the Taylor developments of the
function
at
(cf Chapter 10 section Taylor series).
If
, then the behavior of the function
at
depends on the derivatives of higher order.
The function
is defined by a trigonometric polynomial, hence it is differentiable over
. We have:
. Now check that this first derivative vanishes at
rd
.
We have:
; then
. Therefore the function
has a minimum at
.Similarly, we see that
vanishes at
rd
; there we have
, i.e.
has a maximum at
.
Of course, the above computations are not sufficient for deciding whether these extrema are absolute or local.
Noah Dana-Picard 2007-12-28