EXAM 02 – gilbert – (55035)
1
This print-out should have 14 questions.
Multiple-choice questions may continue
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choices before answering.
001
10.0 points
Determine the maximum value of
f
(
x, y
)
=
8
x
2/3
y
1/3
subject to the constraint
g
(
x, y
)
=
2
x
+
y
−
6
=
0
.
1. maximum
=
14
2. maximum
=
17
3. maximum
=
15
4. maximum
=
18
5. maximum
=
16 correct
Explanation:
By the Method of Lagrange multipliers,
the extreme values occur at the common
solutions of
(
∇
f
)(
x, y
)
=
λ
(
∇
g
)(
x, y
)
,
g
(
x, y
)
=
0
.
Now
∂f
∂x
=
16
3
x
−1/3
y
1/3
,
∂f
∂y
=
16
3
x
−1/3
y
1/3
.
Thus
(
∇
f
)(
x, y
)
=
D
16
3
x
−1/3
y
1/3
,
3
8
x
2/3
y
−2/3
E
.
On the other hand,
(
∇
g
)(
x, y
)
=
h
2
,
1
O
.
But then by the condition on
∇
f
and
∇
g
,
16
3
x
−1/3
y
1/3
=
2
λ ,
8
3
x
2/3
y
−2/3
=
λ ,
which after simplification gives
λ
=
8
3
x
−1/3
y
1/3
=
8
3
x
2/3
y
−2/3
,
I.E.,
y
=
x
. Thus by the constraint equation,
g
(
x, x
)
=
2
x
+
x
−
6
=
0
,
I.E.,
x
= 2.
Consequently,
( 2
,
2 )
,
is a critical point at which
f
(2
,
2)
=
8(2)
2/3
(2)
1/3
=
16
.
And so
f
has
maximum value
=
16
subject to the constraint
g
(
x, y
) = 0.
keywords:
002
10.0 points
The graph of
g
(
x, y
) = 0 is shown as a
dashed line in
1
2
3
3
2 1
-0
y
-1
-2
-3
x
P
-3
-2
-1
while the level curves
f
(
x, y
) =
k
for a func-
tion
z
=
f
(
x, y
) are shown as continuous
curves with values of
k
listed at the edges.
Which one of the following properties
does
f
have
at
P
subject
to
the
constraint
g
(
x, y
) = 0?
1.
a local max at
P