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 has a maximum at .
- If , then has a minimum at .

If , then the behavior of the function at depends on the derivatives of higher order.

- i.
- If even and , then has an maximum at .
- ii.
- If even and , then has an minimum at .
- iii.
- If odd, then has a point of inflexion at .

- Let . Then , . Both derivatives vanish at 0. The first occurrence of a derivative which does not vanish at 0 is the derivative and . It follows that has a minimum at 0 (see Figure 13(a)).
- Let . Then , . Both derivatives vanish at 0. The first occurrence of a derivative which does not vanish at 0 is the derivative. This order is odd, hence has a point of inflexion at 0 (see Figure 13(b)).

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