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Lecture 10  Wednesday April 22nd
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10.1 Constrained Optimization
A set
U
⊂
R
n
is called
closed
if the complement
R
n
\
U
is open. For example, the set
{
(
x,y
)
∈
R
2
:
x
2
+
y
2
≤
1
}
is a closed set since
{
(
x,y
)
∈
R
2
:
x
2
+
y
2
>
1
}
– its
complement – is open. Let
f
:
R
n
→
R
be a function and let
U
be a closed subset of the
domain of
f
so that
f
is continuous on
U
. In this section we use the second derivative
methods we have so far developed to ﬁnd global maxima and minima of
f
on
U
.
Deﬁnition.
Let
f
:
R
n
→
R
be a function and let
U
be a subset of the domain
of
f
. A point
x
∈
U
is a global or absolute maximum of
f
on
U
if
f
(
x
)
≥
f
(
y
) for all
y
∈
U
, and a global or absolute minimum of
f
on
U
if
f
(
x
)
≤
f
(
y
).
The following theorem guarantees the existence of absolute minima and maxima on closed
sets.
Theorem.
Let
f
:
R
n
→
R
be a function and let
U
be a closed subset of the domain
of
f
so that
f
is continuous on
U
. Then
f
has a global maximum and
a global minimum on
U
.
The global extremes guaranteed by this theorem need not be unique –
f
may have many
global maxima and minima on
U
. With the assumption
f
∈
C
2
(
U
), we can use the second
derivative test on
U
to ﬁnd these global extremes.
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.
 Spring '07
 Enright
 Math

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