Extremal points of a function.

Definition 2.5.6   Let $f$ be a function defined on a domain $\mathcal{D}$ ; let $x_0 \in \mathcal{D}$ .

1. The function $f$ has an absolute maximum at $x_0$ if $\forall x \in \mathcal{D}, \; f(x) \leq f(x_0)$ .
2. The function $f$ has a relative (or local) maximum at $x_0$ if there exists an interval $I \subset \mathcal{D}$ such that $\forall x \in I, \; f(x) \leq f(x_0)$ .
3. The function $f$ has an absolute minimum at $x_0$ if $\forall x \in \mathcal{D}, \; f(x) \geq f(x_0)$ .
4. The function $f$ has a relative (or local) minimum at $x_0$ if there exists an interval $I \subset \mathcal{D}$ such that $\forall x \in I, \; f(x) \geq f(x_0)$ .
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 $f$ such that $f(x)=\frac 1x$ has no extremum (see Figure 6(b)).
3. The sine function has an absolute maximum at every point $\pi /2 + 2k \pi$ and an absolute minimum at every point $\-\pi /2 + 2k \pi$ (in both cases, $k$ denotes an integer); see Figure 6(c).

   Figure 6:

   

4. The function $f$ such that $f(x)=x+2\; \vert x-1\vert - \vert 2x+5\vert$ has a local maximum at $-5/2$ and a local minimum at $1$ ; 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.

   Figure 7: The graph of a function with two local extrema.