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Unformatted text preview: p = ( p 1 , p 2 , p 3 ) = (1 , 1 , 0) . Consider the following constrained maximization problem: Max f ( x 1 , x 2 , x 3 ) subject to p 1 x 1 + p 2 x 2 + p 3 x 3 ≤ 1 and ( x 1 , x 2 , x 3 ) ≥ ⎫ ⎬ ⎭ ( M ) Use the extension of Weierstrass theorem to show that problem ( M ) has a solution. 3. [Implicit Function Theorem: (i) = 15 points; (ii) = 15 points] Let A be an m × n matrix and B be a nonsingular m × m matrix. Let F : R n × R m → R m be de f ned by: F ( x, y ) = Ax − By for all x ∈ R n , y ∈ R m (i) Use the implicit function theorem to show that there is an open set X containing n and an open set Y containing m and a unique function g : X → Y such that: (a) F ( x, g ( x )) = 0 for all x ∈ X (b) g (0 n ) = 0 m . (ii) Using (i), show that g ( x ) = B − 1 Ax for all x ∈ X. 2...
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 Fall '08
 STAFF
 Economics, 1 Hour, 15 minutes, 0m, weierstrass theorem, Quadratic form, 617 Second

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