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204 Midterm 2003 - University of California Berkeley...

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University of California, Berkeley Economics 204–First Midterm Test Tuesday August 25, 2003; 6-9pm Each question is worth 20% of the total Please use separate bluebooks for Parts I and II Part I 1. Prove that n 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 first order conditions for a local maximum or minimum of f . Find the unique ( x , y ) at which the first order conditions are satisfied. (b) Find 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). Find 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 2 ) 2 Use this information to determine whether f has a local max- imum, a local minimum, or neither, at ( x , y ) and to describe the level sets of f
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