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Solution to Problem Set 4 November 9 2013 f x x3 12x f cid 48 x 3x2 12 0 x 2 or 2 f x y x2 2xy y3 y f1 2x 2y 0 f2 2x 3y2 1 0 x 1 y 1 or x y 1 3 1 3 1 i ii 2 i f x x3 3x f cid 48 x 3x2 3 0 x 1 or 1 f cid 48 cid 48 x 6x For x 1 f cid 48 cid 48 1 0 local minimum For x 1 f cid 48 cid 48 1 0 local maximum ii f x1 x2 x2 1 x3 2 2x1x2 f1 2x1 2x2 0 f2 3x2 2 2x1 0 2 3 x1 0 x2 0 or x1 x2 2 3 f11 2 0 f22 6x2 f12 2 For x1 x2 0 f11f22 0 cid 54 f 2 3 x2 2 For x1 2 3 Lagrangian function 12 no conclusion 3 f11f22 2 4 8 4 f 2 12 local minimum L x2 1 x2 3x2 2 x1 2x2 5 First order conditions L x1 2x1 0 1 L x2 1 6x2 2 0 x1 2x2 5 0 x1 2 x2 4 3 2 2
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