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204Final082107

# 204Final082107 - Economics 204Final E 9am-12pm Each of the...

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Economics 204–Final Exam–August 21, 2007, 9am-12pm Each of the four questions is worth 25% of the total Please use three separate bluebooks, one for each of the three Parts Part I 1. State and prove the Supremum Property. 2. Prove that if a finite set X has exactly n elements, then 2 X , the set of all subsets of X , has exactly 2 n elements. Hint : use induction. Part II 3. Consider the function f ( x, y ) = x 3 y 3 + 2 x 2 + 2 xy + 8 y 2 9 x 15 y 2 (a) Compute the first order conditions for a local maximum or minimum of f . Show that the first order conditions are satisfied at the point ( x 0 , y 0 ) = (1 , 1). (b) Compute D 2 f ( x 0 , y 0 ) and give the quadratic Taylor polynomial for f at the point ( x 0 , y 0 ). (c) Find the eigenvalues of D 2 f ( x 0 , y 0 ) and determine whether f has a local max, a local min, or a saddle at ( x 0 , y 0 ). (d) Does f have a global max, a global min, or neither, at ( x 0 , y 0 )? (e) Find the eigenvectors of D 2 f ( x 0 , y 0 ) and provide an orthonormal basis for R 2 consisting of eigenvectors. Rewrite the quadratic Taylor polynomial for f at the point ( x 0 , y 0 ) in terms of this basis.
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