O
ptimization
Now that we have discussed local maximums and minimums of functions, it is natural
to consider applying these concepts to actual problems. In addition to just finding a local
maximum or minimum, we might want to find the largest or smallest value on a region
R
.
These values are called the
global maximum
and
global minimum
of the function on
the region
R
. That is, we have
Global Maximums and Global Minimums of Functions
f
(
x
,
y
) has a global maximum on a region
R
at the point (
x
0
,
y
0
)
provided
f
(
x
0
,
y
0
)
¥
f
(
x
,
y
) for all points (
x
,
y
) in
R
f
(
x
,
y
) has a global minimum on a region
R
at the point (
x
0
,
y
0
) provided
f
(
x
0
,
y
0
)
§
f
(
x
,
y
) for all points (
x
,
y
) in
R
A natural question is whether or not a function has a global maximum or global
minimum value on a region
R
. Thinking back to one-variable calculus, the function
y
=
x
2
has a global minimum at (0, 0), but goes off to infinity as the
x
-values get further away
from 0. But if we specify a set of
x
-values, such as [-1, 2], then we can say that the
function has a global maximum on [-1, 2]. In particular, the maximum occurs at
x
= 2.
There was a result in one-variable calculus that showed that a continuous function (of
one-variable) has a global maximum and a global minimum on [
a
,
b
] for constants
a
and
b
. An analogous result holds for functions of two or more variables.
Extreme Value Theorem for Multivariable Functions
If
f
(
x
,
y
) is a continuous function on a closed and bounded region
R
,
then
f
(
x
,
y
) has a global maximum at some point (
x
1
,
y
1
) in
R
and a global
minimum at some points (
x
2
,
y
2
) in
R
.
A region
R
is said to be closed if the region contains is boundary. (In terms of a graph
of the region, the region is outlined with solid lines as compared to dashed lines. This is
analogous to requiring an interval [
a
,
b
] in one-variable calculus as opposed to (
a
,
b
),
which did not include the end points.) A region is bounded if the values of the functions
are finite (not infinity).
1