## Extremal points of a function.

Definition 2.5.6   Let be a function defined on a domain ; let .
1. The function has an absolute maximum at if .
2. The function has a relative (or local) maximum at if there exists an interval such that .
3. The function has an absolute minimum at if .
4. The function has a relative (or local) minimum at if there exists an interval such that .
5. A maximum and a minimum are called extremal points (singular: extremum

Example 2.5.7

1. The absolute value function has an absolute maximum at 0, but has no minimum (see Figure 6(a)).
2. The function such that has no extremum (see Figure 6(b)).
3. The sine function has an absolute maximum at every point and an absolute minimum at every point (in both cases, denotes an integer); see Figure 6(c).

4. The function such that has a local maximum at and a local minimum at ; see Figure 7. In order to draw the graph, the reader has to first draw a table, as in Chapter 1, section section absolute value; as the function is affine by parts'', the drawing process is easy.

Noah Dana-Picard 2007-12-28