Problem_Set_8 - T. Mitra Fall, 2008 Economics 6170 Problem...

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Unformatted text preview: T. Mitra Fall, 2008 Economics 6170 Problem Set 8 [Due on Wednesday, November 19] 1. [Intermediate Value Theorem] Here is a statement of the Intermediate Value Theorem for continuous real valued functions of a real variable. Theorem: Let f be a continuous real valued function on the closed interval A = [ a,b ] . Suppose x,y A satisfy f ( x ) > f ( y ) . Then for every c, satisfying f ( x ) > c > f ( y ) , there is z A, such that f ( z ) = c. We want to use this theorem to prove the following version of the Inter- mediate Value Theorem for continuous real valued functions of several real variables. Corollary: Let B be a convex subset of R n , and let F be a continuous real valued function on B. Suppose x 1 ,x 2 B satisfy F ( x 1 ) > F ( x 2 ) . Then for every c, satisfying F ( x 1 ) > c > F ( x 2 ) , there is v B, such that F ( v ) = c. Proceed with the following steps. (a) Define A = [0 , 1] , and for each t A, define f ( t ) = F ( tx 1 +(1- t ) x 2 ) ....
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Problem_Set_8 - T. Mitra Fall, 2008 Economics 6170 Problem...

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