Question

Solve the following problem using Lagrange multiplier method: Maximize f(x,y,z) = 4y-2z subject to the constraints 2x-y-z 2 x2+y2- 1 1. (1) (2) (Note: You need not check the Hessian matrix, just find the maximum by evaluating the values of f(x,y,z) at the potential solution points) Also, using sensitivity analysis, find the change in the maximum value of the function, f, if the above constraints are changed to: (3) (4) 2x -0.9y-z 2 x2+ y2- 0.9.

Answer 1

Ang:- iven dota, 4gg tagrange mulkpler method is Martmeze Flx,9,2) uy-22 Sulject to the isag braints 2メーy-2=2 L2) he Eguation he Lagrang 's Lomas x,y,2, ,- Clay, 2) +, 9,lx,y,2)+232) where and Lagrange 's MulRples 入 core 2X-y-2-2 3, (x,y,2) 9 (xy,2) L (,2,,) y-22 t(2x-y-2-2)+ (-) New, 2, +2 , o- Ci) )L

42y o i) Also, JL O -2-,0 () Also iv) 22-y-2-2 =0 Also, JL 2 3olumg (i to ( uatsen puttng 2 09n 2 -u+2x , o Oh, X Also, A lso. ナ9 2 -I O

091 13- 3 t 3.606 2 Por 3.60 0-554 3:606 - 3 3.606 Por O.832 3-606 Atso Ror >ミ-3-606 ond y o.832 X: -0-SS 2. (0.5su)-1-0 832)-20.66 AISO 92 (-0.55u) -(o.32)-2 -3-94 (,9,2, lo.ssu,-0.832,0.06,-2, 3-606) and (-0.554, 0.832,-39,-236o6) (0.832)-2o.06)3.20 P(X,y, 2) (o.832)-2-23.9u) - l1-208 uメ ond 11.208 at Pl7,,2) Marmur value -0.55u, 0.832, -3.9u)

gengituithy onalysis NOw da,- (0.4-1) -o-832 x +0-0832 AIso, -0.9) +0.1 -7 92 09n -(-2) (0.0832) - 13-6e6) (0.1) 4s.o CS

Answer 2