746
CHAPTER 15
DIFFERENTIATION IN SEVERAL VARIABLES
(ET CHAPTER 14)
27.
f
(
x
,
y
)
=
2
x
−
y
,0
≤
x
≤
1
,
0
≤
y
≤
3
SOLUTION
f
is maximum when
x
is maximum and
y
is minimum, that is
x
=
1and
y
=
0.
f
is minimum when
x
is minimum and
y
is maximum, that is,
x
=
0,
y
=
3. Therefore, the global maximum of
f
in the set is
f
(
1
,
0
)
=
2
·
1
−
0
=
2 and the global minimum is
f
(
0
,
3
)
=
2
·
0
−
3
=−
3.
28.
f
(
x
,
y
)
=
(
x
2
+
y
2
+
1
)
−
1
≤
x
≤
3
,
0
≤
y
≤
5
f
(
x
,
y
)
=
1
x
2
+
y
2
+
1
is maximum when
x
2
and
y
2
are minimum, that is, when
x
=
y
=
0.
f
is minimum
when
x
2
and
y
2
are maximum, that is, when
x
=
3and
y
=
5. Therefore, the global maximum of
f
on the given set is
f
(
0
,
0
)
=
(
0
2
+
0
2
+
1
)
−
1
=
1, and the global minimum is
f
(
3
,
5
)
=
(
3
2
+
5
2
+
1
)
−
1
=
1
35
.
29.
f
(
x
,
y
)
=
e
−
x
2
−
y
2
,
x
2
+
y
2
≤
1
The function
f
(
x
,
y
)
=
e
−
(
x
2
+
y
2
)
=
1
e
x
2
+
y
2
is maximum when
e
x
2
+
y
2
is minimum, that is, when
x
2
+
y
2
is minimum. The minimum value of
x
2
+
y
2
on the given set is zero, obtained at
x
=
0and
y
=
0. We conclude that the
maximum value of
f
on the given set is
f
(
0
,
0
)
=
e
−
0
2
−
0
2
=
e
0
=
1
f
is minimum when
x
2
+
y
2
is maximum, that is, when
x
2
+
y
2
=
1. Thus, the minimum value of
f
on the given disk
is obtained on the boundary of the disk, and it is
e
−
1
=
1
e
.
30. Assumptions Matter
Show that
f
(
x
,
y
)
=
x
+
y
has no global minimum or maximum on the domain 0
<
x
,
y
<
1. Does this contradict Theorem 3?
The largest and smallest values of
f
on the closed square 0
≤
x
,
y
≤
1are
f
(
1
,
1
)
=
2and
f
(
−
1
,
−
1
)
=
−
2. However, on the open square 0
<
x
,
y
<
1,
f
can never attain these maximum and minimum values, since the
boundary (and in particular the points
(
1
,
1
)
and
(
−
1
,
−
1
)
) are not included in the domain. This does not contradict
Theorem 3 since the domain is open.
31.
Let
D
={
(
x
,
y
)
:
x
>
0
,
y
>
0
}
. Show that
D
is not closed. Find a continuous function that does not have a global
minimum value on
D
.
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 Fall '09
 Park
 Critical Point, Multivariable Calculus, F E R E N T, S E V E R A L VA R

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