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204Final081908

# 204Final081908 - Economics 204Final E 9am-12pm Each of the...

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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 X and Y are vector spaces over the same field F and dim X = dim Y , then X and Y are isomorphic. 2. Consider the function f ( x, y ) = 3 x 2 + 3 y 2 2 xy + x 4 + y 5 (a) Show that 0 0 is a critical point of f . (b) Determine whether f has a local max, a local min, or neither at 0 0 . (c) Does f have 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 y 1 = y and y 2 = 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 y 1 y 2 = y 2 y 3 1 / 100 y 1 y 3 2 / 100 , y 1 (0) = y 2 (0) = 1 (2) Show that the unique stationary point for Equation (2) is 0 0 . Show that the

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204Final081908 - Economics 204Final E 9am-12pm Each of the...

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