204Final081908 - Economics 204Final ExamAugust 19, 2008,...

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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 Feld F and dim X =d im 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 0 = y, y (0) = y 0 (0) = 1 (1) (a) Write this as a Frst order linear Initial Value Problem using the variables y 1 = y and y 2 = y 0 . (b) ±ind the eigenvalues of the matrix obtained in part (a). (c) ±ind 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.
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.

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204Final081908 - Economics 204Final ExamAugust 19, 2008,...

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