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