1. (i) Prove the converse of the concavity characterisation: For any x,y e X and 1 € (0,1) we have f(4x + (1 - 1)y) 2 1f

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1 I Prove The Converse Of The Concavity Characterisation For Any X Y E X And 1 0 1 We Have F 4x 1 1 Y 2 1f 1 (249.32 KiB) Viewed 15 times

1. (i) Prove the converse of the concavity characterisation: For any x,y e X and 1 € (0,1) we have f(4x + (1 - 1)y) 2 1f (x) + (1 - 1)f(y) = f is concave (ii) Consider the Cobb-Douglas production function f(K,L) = AK“LB. What conditions on A, a and ß do you require for the function to be concave? 2. (20 PTS) (i) (8 PTS) Find whether the function f(x,y) = (3 - x - y)xy has a local maxima or a minima or neither. (ii) (8 Pts) Calculate the 2nd order Taylor expansion of the function f(x,y) = x2 + y2 + xy around the point (1,2). (iii) (4 pts) Prove the following assertion: If f is convex, then -f is concave.