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University of California, Berkeley
Economics 204–First Midterm Test
Tuesday August 25, 2003; 69pm
Each question is worth 20% of the total
Please use separate bluebooks for Parts I and II
Part I
1. Prove that
n
X
k
=1
k
=
n
(
n
+1)
2
2. Consider the function
f
(
x, y
)=
e
x
2
−
6
xy
+
y
2
Recall that
d
dz
e
z
=
e
z
.
(a) Compute the Frst order conditions for a local maximum or
minimum of
f
. ±ind the unique (
x
∗
,y
∗
)a
twh
ichtheF
r
s
t
order conditions are satisFed.
(b) ±ind the second order Taylor series expansion of
f
at the
point (
x
∗
,y
∗
) determined in part (a); your answer should in
volve a symmetric matrix
A
representing the quadratic terms
in the expansion.
(c) Diagonalize the matrix
A
you found in part (b). ±ind an or
thonormal basis
{
v
1
,v
2
}
of
R
2
such that the quadratic terms
of the Taylor expansion can be written as
g
((
x
∗
,y
∗
)+
γ
1
v
1
+
γ
2
v
2
)=
λ
1
(
γ
1
)
2
+
λ
2
(
γ
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.
 Fall '08
 ANDERSON
 Economics

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