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property? The purpose of the present section is to investigate whether or not this is
the case.
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In this section we will need some of the basic properties of the maxima of
functions defined on closed sets. These results are, for the sake of easy reference
and completeness, recaptured without any derivations or proofs. The reader unfa-
miliar with these concepts and properties should consult a suitable introductory
mathematics text.
Let us first consider functions of a single variable x. Suppose q.x/ is a smooth
function defined on the closed unit interval Œ0; 1. Assume that q achieves its
maximum value at an interior point
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x
, i.e.,
x
2
.0; 1/
and q.x/
q.x
/ for all x
2
Œ0; 1;
then q must satisfy
q
0
.x
/
D
0;
(8.19)
q
00
.x
/
0;
(8.20)
see any introductory text on calculus. Consequently, if a smooth function h.x/,
defined on Œ0; 1, is such that
h
0
.x/
¤
0
for all x
2
.0; 1/
or
h
00
.x/ > 0 for all x
2
.0; 1/;
then h must achieve
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its maximum value at one of the endpoints of the unit interval.
Hence, we conclude that
h.0/
h.x/
for all x
2
Œ0; 1
(8.21)
or
h.1/
h.x/
for all x
2
Œ0; 1;
(8.22)
must hold. This means that
h.x/
max .h.0/; h.1//
for all x
2
Œ0; 1:
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From a modeling point of view this is an important question. High-quality models must, of
course, fulfill the basic properties of the underlying physical process!
11
Any point p in the open interval .0; 1/ is referred to as an interior point of the closed interval
Œ0; 1.
12
Recall that a smooth function defined on a closed interval is bounded. Furthermore, such a
function will always achieve both its maximum and minimum value within this interval, see e.g.,
Apostol [4].
