Application to finding extremal points.

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).

Proposition 6.10.11   Let be a function defined on a neighborhood of . Suppose that is differentiable (at least) twice at and that .
1. If , then has a maximum at .
2. If , then has a minimum at .

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

Proposition 6.10.12   Let be a function defined on a neighborhood of . Suppose that has enough derivatives on this neighborhood and that . Denote by the order of the first derivative which does not vanish at .
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 .

Example 6.10.13

1. 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)).
2. 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)).

Example 6.10.14   Let ; the graph of is displayed on Figure 14.

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