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1、Chapter2/TheMathematicsofOptimizationv5CHAPTER2THEMATHEMATICSOFOPTIMIZATIONTheproblemsinthischapterareprimarilymathematical.TheyareintendedtogivestudentssomepracticewithtakingderivativesandusingtheLagrangiantechniques,buttheproblemsinthemselvesofferfeweconomi
2、cinsights.Consequently,nocommentaryisprovided.Alloftheproblemsarerelativelysimpleandinstructorsmightchoosefromamongthemonthebasisofhowtheywishtoapproachtheteachingoftheoptimizationmethodsinclass.Solutions2.1a.b.8,12c.d.e.f.g.U=16contourlineisanellipsecentered
3、attheorigin.Withequation,slopeofthelineat(x,y)is.2.2a.ProfitsaregivenbyChapter2/TheMathematicsofOptimizationv5b.soprofitsaremaximizedc.soq*=10obeysMR=MC=50.2.3Substitution:Note:.Thisisalocalandglobalmaximum.LagrangianMethod:so,x=y.usingtheconstraintgives2.4Se
4、ttinguptheLagrangian:.Chapter2/TheMathematicsofOptimizationv5So,x=y.Usingtheconstraintgives.2.5a..b.Substitutingfort*,..c.dependsongbecauset*dependsong.so.d.,areductionof.08.Noticethatsoa0.1increaseingcouldbepredictedtoreduceheightby0.08fromtheenvelopetheorem
5、.2.6a.Thisisthevolumeofarectangularsolidmadefromapieceofmetalwhichisxby3xwiththedefinedcornersquaresremoved.b..Applyingthequadraticformulatothisexpressionyields.Todeterminetruemaximummustlookatsecondderivative--whichisnegativeonlyforthefirstsolution.Chapter2/
6、TheMathematicsofOptimizationv5b.IfsoVincreaseswithoutlimit.c.ThiswouldrequireasolutionusingtheLagrangianmethod.Theoptimalsolutionrequiressolvingthreenon-linearsimultaneousequations—atasknotundertakenhere.Butitseemsclearthatthesolutionwouldinvolveadifferentrel
7、ationshipbetweentandxthaninpartsa-c.2.7a.SetupLagrangianyieldsthefirstorderconditions:Hence,.Withk=10,optimalsolutionisb.Withk=4,solvingthefirstorderconditionsyieldsc.OptimalsolutionisAnypositivevalueforx1reducesy.d.Ifk=20,optimalsolutionisBecausex2providesad
8、iminishingmarginalincrementtoywhereasx1doesnot,alloptimalsolutionsrequirethat,oncex2reaches5,anyextraamountsbedevotedentirelytox1.2.8Theproofismosteasilyaccomplishedthroughtheuseofthematr
