ex maximize 36 x2 y subject to the constraint Hx g 79 25 91 9 7g 25 0 36 x2 y 36 x2 y F x y A 1 79 25 29 7 0 Ex 79 25 0 the 1 two equations for 2 7 0 2x 2g 77 x 0 9 0 equate the two equations for y 2X Ey or Substitute this expression for in equation If above 25 0 25 y 7y y 5 y y Ey E Back substitute to find X and X If If f max value occurs at 36 115 1312 410 289 subject to the constraint cont of ex in 2 5 289 using Lagrange multipliers minimize 42 00 xy o where x andy are restricted to positive values IXg 42 F x y 1 Txy o 1 4 2 IE 42 Xy o Fy 28 Xx o IF 600 Xy 0 91 197 600 XY 1600 9 28 xx 289 42 o o 424 289 y y 2 38 30 20 600 13g y 592 600 31600 900 y g 130 y esP oniyo the minimum value of 42 289 the constraint subject to ay w occurs when 20 9 30 Minimum value 42120 28130 16800 units of labor and y units of capital ex suppose 60 y units of a certain product Also each roduce fix y hit of labor costs 100 whereas each unit of capital costs 200 ssume 30 000 is available to spend on production How many hits of labor and capital should be utilized to maximize production let must use all money 30 000 can units of labor units of capital y no g x y 30 000 100 2009 0 Objective fix g 60 y y 4 y 19 I y 01 x y j y 09 y 225 37 5 Y y i themax X 30 000 loux 2009 3 60 y y i Fix y d 2 45 If IS IF 30 000 100 y 1001 o 2007 0 2004 0 30 000 100 100X Eu 4 200 tx O 28 301000 301000 Eo 112250 production is achieved by using 225 units oflabort37 sunitsoteapitatoo fact IE laborand capital are at their optimal levels the ratio the ratio of their unit cost of their marginal productivities Eggnogfan If doing Lagrange multipliers in 3 variables F x y Z X f x y Z dg x y Z If o Fo Ez o TE o to the constraint 92 1478 F x y Z X Fx Ily 152 XYZ lixy 1492 15 2 If 11 147 840 Xyz 142 1 2 0 FE 14g Isx Xxy IF 47840 Xyz Xyz 119 152 11115 Xxy 149 15 144T I 15 hI I H i y IyH IF 47840 XYZ O 147840 144121 5 147 4 g Two 60 15 216000 160 56 60 44 2 t t