The function
has a relative (or local) maximum at
if there exists an interval
such that
.
The function
has an absolute minimum at
if
.
The function
has a relative (or local) minimum at
if there exists an interval
such that
.
A maximum and a minimum are called extremal points (singular: extremum
Example 2.5.7
The absolute value function has an absolute maximum at 0, but has no minimum (see Figure 6(a)).
The function
such that
has no extremum (see Figure 6(b)).
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).
Figure 6:
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.
Figure 7:
The graph of a function with two local extrema.