Answer:
Let (
x
n
,y
n
,z
n
)
∈
D
with (
x
n
,y
n
,z
n
)
→
(
x,y,z
). Since
z
n
= 0,
z
= 0. Because
C
is closed
and (
x
n
,y
n
)
∈
C
and (
x
n
,y
n
)
→
(
x,y
), we ﬁnd (
x,y
)
∈
C
, so (
x,y,z
) = (
x,y,
0)
∈
D
. Thus
D
is closed because it contains all of its limit points.
4. Let
f
(
x,y,z
) =
x
2
+ 3
y
+
z
3

5.
a
) Find an (
x
0
,y
0
,z
0
) satisfying
f
(
x
0
,y
0
,z
0
) = 0.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentMATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 6, 2000
Page 2
Answer:
The point (1
,
1
,
1) works.
b
) Can
x
be expressed as a function
g
(
y,z
) in some neighborhood of (
x
0
,y
0
,z
0
)?
Answer:
Since
∂f/∂x
= 2
x
,
∂f/∂x
= 2 at (
x
0
,y
0
,z
0
). The Implicit Function Theorem yields
such a function
g
. Alternatively, note that
g
(
y,z
) = (5

3
y

z
3
)
1
/
2
works.
c
) Compute
dg
.
Answer:
By the Implicit Function Theorem,
dg
= (

1
/
2
x
)(
∂f/∂y,∂f/∂z
) =

(3
/
2)(1
,
3
z
2
).
At (1
,
1
,
1), this has the value (

3
/
2
,

3
/
2).
5. Consider the function
u
(
x,y
) =
x
+
√
y
.
a
) Does
u
attain a maximum on the set
A
=
{
(
x,y
) :
x,y
≥
0,
px
+
y
≤
3
}
? Why?
Answer:
The function
u
is continuous. The set
A
is a budget set in
R
2
with strictly positive
prices. We saw in class that such a set is compact. The Weierstrass Theorem says that any
continuous function attains a maximum on a compact set.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 STAFF
 Economics, Topology, Metric space, λ, Closed set, mathematical economics midterm

Click to edit the document details