01test2 - Mathematical Economics Midterm#2 November 8 2001...

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Mathematical Economics Midterm #2, November 8, 2001 1. Consider the sequence x n = 2 n + ( - 2) n + 1 /n 2 . a ) Does the sequence converge? If so, what is its limit? Answer: The sequence does not converge. In fact, | x n - x n +1 | > 2 n , so the terms get farther apart. b ) If the sequence doesn’t converge, does it have any convergent subsequences? If so, identify one of them and compute its limit. Answer: Yes, it has convergent subsequences. One such subsequence is the subsequence of odd terms, x n j = x 2 j +1 = 1 / (2 j + 1) 2 . This subsequence converges to 0. 2. Find all local and global maxima and minima of the function f ( x, y ) = (2 / 3) x 3 + x 2 + 2 y 2 - 2 xy + 4 x - 10 y . Answer: First compute df = (2 x 2 + 2 x - 2 y + 4 , 4 y - 2 x - 10). Setting df = (0 , 0), we find 2 y = x + 5 from the second equation. Substituting in the first equation, this implies 2 x 2 + x - 1 = 0. This has solutions x = - 1 and x = 1 / 2. It follows that the critical points are ( - 1 , 2) and (1 / 2 , 11 / 4). We next compute the Hessian: d 2 f = 4 x + 2 - 2 - 2 4 . At ( - 1 , 2), we obtain H 1 = - 2 < 0 and H 2 = - 8 - 4 = - 12 < 0. The Hessian is indefinite and ( - 1 , 2) is neither a local minimum nor local maximum. At (1 / 2 , 11 / 4), we obtain H 1 = 4 > 0 and H 2 = 16 - 4 = 12 > 0. The Hessian is now positive definite and (1 / 2 , 11 / 4) is a local minimum.
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