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Unformatted text preview: R + , deﬁned by: f ( x ) = ax + b [ x/ (1 + x )] for all x ≥ Consider the following constrained maximization problem: Maximize f ( x )x subject to x ≥ ± ( P ) (a) If a < 1 , show that there exists a solution to problem ( P ) . (b) If a ≥ 1 , show that there is no solution to problem ( P ) . 5. [Extension of Weierstrass Theorem] Let p and q be arbitrary positive numbers, and let f : R 2 + → R be a continuous function on R 2 + . Suppose there is (¯ x 1 , ¯ x 2 ) ∈ R 2 + which satisﬁes f (¯ x 1 , ¯ x 2 ) = 1 . Consider the constrained minimization problem: Minimize px 1 + qx 2 subject to f ( x 1 ,x 2 ) ≥ 1 and ( x 1 ,x 2 ) ∈ R 2 + ( Q ) Show that there is a solution to problem ( Q ) . 2...
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This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell.
 Fall '08
 MITRA

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