Problem_Set_9 - T.Mitra, Fall, 2008 Economics 6170 Problem...

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Unformatted text preview: T.Mitra, Fall, 2008 Economics 6170 Problem Set 9 (Due on Monday, December 1) 1. [Unconstrained Optimization: First-Order Conditions] Here is a statement of the first-order condition for a maximum of real valued functions of a real variable. Theorem: Let f be a continuously differentiable real valued function on the interval A = ( a,b ) . If c A satisfies f ( c ) f ( t ) for all t A, then f ( c ) = 0 . We want to use this theorem to prove the following version of the first- order condition for a local maximum of real valued functions of several real variables. Corollary: Let C be an open subset of R n , and let F be a continuously differentiable real valued function on C. If x C is a point of local maximum of F, then F ( x ) = 0 . Proceed with the following steps. (a) Since x C is a point of local maximum of F, we can find r > 0 such that B B ( x, 2 r ) C, and F ( x ) F ( x ) for all x B. (b) Pick any k { 1 ,...,n } , and define a ( k ) = x- re k ,b ( k ) = x + re k , where e k is the k-th unit vector in R n . Then, by definition of B, we have...
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Problem_Set_9 - T.Mitra, Fall, 2008 Economics 6170 Problem...

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