MATH 2853 SUPPLEMENTAL HOMEWORK PROBLEMS
4.2 Extrema
1
Find all critical points and determine whether they are local maxima, local minima, or neither, for
the following functions:
(i)
f
(
x
,
y
,
z
)
=
x
2

4
y
2
+
z
2
+
8
yz
+
8
xy
, and
(ii)
f
(
x
,
y
,
z
)
=
x
2
+
y
2

z
3
+
3
z
.
(iii)
f
(
x
,
y
,
z
)
=
x
4
+
x
2

6
xy
+
3
y
2
+
z
2
.
2
A firm has a monopoly on two products and can set the prices. Assume the demand for the products
by consumers is
Q
1
=
145

2
P
1

P
2
and
Q
2
=
160

P
1

3
P
2
. Assume that the cost is given
by
C
=
5
Q
1
+
5
Q
2
, and the profit is given by
π
=
P
1
Q
1
+
P
2
Q
2

C
or
π
=
P
1
(
145

2
P
1

P
2
)
+
P
2
(
160

P
1

3
P
2
)

5
(
145

2
P
1

P
2
)

5
(
160

P
1

3
P
2
).
What are the prices that maximize profit? Use the second derivative test to show that these prices
give a point that locally maximizes the profit.
3
A company operates two plants which manufacture the same item and whose cost functions are
C
1
=
8
.
5
+
0
.
03
Q
2
1
and
C
2
=
5
.
2
+
0
.
04
Q
2
2
,
where
Q
1
and
Q
2
are the quantities produced by each plant, and
Q
=
Q
1
+
Q
2
is the total quantity
produced. The prices are determined by the total amount produced by
P
=
60

0
.
04
Q
.
How much should each plant produce in order to maximize the company’s profit?
Hint: The profit is
π
=
P Q

C
1

C
2
=
(
60

0
.
04
Q
)
Q

C
1

C
2
=
60
Q

0
.
04
Q
2

C
1

C
2
=
60
(
Q
1
+
Q
2
)

0
.
04
(
Q
1
+
Q
2
)
2

8
.
5

0
.
03
Q
2
1

5
.
2

0
.
04
Q
2
2
.
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 Spring '08
 Rogerson
 Math, Critical Point, 6W, Stokes Theorems, SUPPLEMENTAL HOMEWORK PROBLEMS

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