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Math E21a – Fall 2009 – Exam #2 solutions
1) Find all critical points of the function
32
(, )
6
6 3
2
fxy x y
x
y x y
and, for each point, determine whether it
is a local maximum, a local minimum, or a saddle point.
Solution
: The critical points (stationary points) occur where both partial derivatives vanish. We calculate:
22
36
6
3
(2
2
)
0
(6
3) 2
6
5
(
1)(
5)
0
263
0
x
y
fxy
xy
xx
x
x
x
x
fy
x
y
x
These equations gives that either
1
x
or
5
x
. If we substitute these individually into the 2nd equation, we get the
two critical points
3
2
(1, ) and
27
2
(5,
) . We then test each of these using the 2
nd
derivative test.
We first calculate the Hessian matrix of 2
nd
derivatives:
66
62
xx
xy
f
yx
yy
ff
x
H
.
For each of the critical points, we calculate:
3
2
(1, )
f
H
,
3
2
det
12 36
24
0
f
H
,so
3
2
(1, ) must be a saddle point.
27
2
30
6
(5,
)
f
H
,
27
2
det
(5,
)
60 36
24
0
f
H
and
30
0
xx
f
, so
27
2
(5,
) must be a local
minimum.
2) Suppose that a student member of a Harvard committee has determined that the wellbeing of a workstudy student is
described by the function
0.6
0.4
Wxy x y
, where
x
is her hourly salary and
y
is the number of hours per week she
spends in class. The Administration has decreed that this committee may only recommend values of
x
and
y
that satisfy
the constraint
24
0
. Use the Method of Lagrange Multipliers to find the values of
x
and
y
that maximize student
wellbeing according to this model.
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 Fall '09
 
 Critical Point

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