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Unformatted text preview: Question 3. Consider the function f ( x 1 ,x 2 ) = x 1 + x 2 + ax 1 x 21 2 x 2 11 2 x 2 2 dened on the Euclidean plane, < 2 . A) What is the gradient of f ? Evaluate this gradient at the point x 1 = 1, x 2 = 2. B) Find the Hessian of f . C) For what values of a , if any, is f a concave function? For what values of a if any is f a convex function? Justify your answer. D) For what values of a does f have a global maximum? If f has a global maximum, nd this maximum. Question 4. A) State Eulers theorem on homogeneous functions. B) Suppose that the function f ( x 1 ,...,x n ) is homogeneous of degree k . Consider the function f i ( x 1 ,...,x n ) = f ( x 1 ,...,x n ) x i . Is this also a homogeneous function? If so, of what degree is it homogeneous? C) Extra Credit: Supply proofs for A and B....
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
 Spring '10
 fallahi

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