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:
    \begin{figure}\centering
\mbox{
\subfigure[Absolute value]{\epsfig{file=Absolute...
...ubfigure[The sine function]{\epsfig{file=sinus.eps,height=4cm}}
}\end{figure}

  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.
    \begin{figure}\centering
\mbox{
\subfigure[]{\epsfig{file=sin1_x-01.eps,height=...
...uad\qquad
\subfigure[]{\epsfig{file=sin1_x-02.eps,height=4cm}}
}\end{figure}

Noah Dana-Picard 2007-12-28