Economics 204–Final Exam–August 19, 2008, 9am-12pm
Each of the four questions is worth 25% of the total
Please use three separate blue/greenbooks, one for each of the three Parts
Part I
1. Prove that if Xand Yare vector spaces over the same field Fand dim X=dimY,
then Xand Yare isomorphic.
2. Consider the function
f(x, y)=3x2+3y2−2xy +x4+y5
(a) Show that 0
0is a critical point of f.
(b) Determine whether fhas a local max, a local min, or neither at 0
0.
(c) Does fhave a global max, a global min, or neither at 0
0?
Part II
3. Consider the Initial Value Problem
y =−y, y(0) = y(0) = 1 (1)
(a) Write this as a first order linear Initial Value Problem using the variables y1=y
and y2=y.
(b) Find the eigenvalues of the matrix obtained in part (a).
(c) Find the unique solution of the Initial Value Problem in Equation (1). Hint:you
can use the product of three complex matrices if you wish, but there is a simpler
approach.
(d) Now consider the Initial Value Problem
y1
y2
=y2−y3
1/100
−y1−y3
2/100 ,y
1(0) = y2(0) = 1 (2)
Show that the unique stationary point for Equation (2) is 0
0. Show that the
linearized equation corresponding to Equation (2) is the Initial Value Problem you
found in part (a). Find a function G:R2→R+such that every solution of the
linearized differential equation follows a level set of G.
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