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Unformatted text preview: Econ 310 Problem Set 1 Lucas Manuelli (&rst question being slightly altered) October 13, 2008 Question 1 (b) For x > we have f ( x ) = 80 & 20 x so f (4) = 0 : In addition for x > we have f 00 ( x ) = & 20 so the function is concave for positive x so that x = 4 is a local max. For x < we have f ( x ) = 2 x & 4 so f ( x ) < and there is no critical point when x < : Sometimes, critical points are also called the ones where derivative doesn¡t exist. We¡ ll deal with those points in the next part. (c) This function is continuous everywhere since x ( x & 4) and 10 x (8 & x ) are continous functions and at the point of intersection they connect properly, i.e. f (0) = lim x ! 0+ f ( x ) = 0 . At x = 0 , derivative does not exist since leftsede derivative is f (0) = & 4 and rightside derivative is f (0) = 80 : Since the derivative changes its sign from negative to positive at x = 0 there is a local minimum....
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 '08
 StephenE.Morris
 Critical Point, Optimization, 4m, 0 m, …rst order conditions

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