MATH 125 Final: MATH 125 UWashington Quiz125FinalA09

• School: UW
• Department: Mathematics
• Course: MATH 125
• Professor: All

Document Summary

Mathematical economics exam #2, november 4, 2013: consider the problem of maximizing x + y subject to the constraint that x2 + y2/4 1, without calculating it, prove this problem has a solution, find the solution. Answer: since the constraint set is compact and x + y is continuous, the weierstrass theorem guarantees there is a solution, form the lagrangian l = x + y (x2 + y2/4 1). The rst-order conditions are 1 = 2 x and. Here > 0, so x2 + y2/4 = 1 by complementary slackness. Now 2x = y/2 so x = y/4 and (y/4)2 + y2/4 = 1. Answer: now u/ x = 2x exp( x2 y2) > 0 and u/ y = 2y exp( x2 y2) > 0 when (x, y) r2. Explain: if you answered (b) af rmatively, call such a function g and nd dg(1/2, 1/2). Answer: set 2 = f(1/2, 1/2, z) = z2/2.