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Unformatted text preview: R + , dened 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 satises 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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- Fall '08