Question

7) Given f(x,y)= x^2+y^2+2, subject to the constraint g(x,y)=x^2+xy+y^2-4=0, write the system of equations which must be solved to optimize f using Lagrange Multipliers.

Answer 1

- Given f(x,y) = x² tý tz subject to the construint g(x,y) = x² + xy + y - 40 0 Construct a Lagrange function 162,5) = f(1,3) +49(0, 3) :!.1,) = (34+y+z)+1(22 +39 +92-4) For critical points off, we solve, i.c. 2x + 2(x + y)=0 – fzy td (x +2y) =0 – We must solve systern of equations , 0, to optimize f using Lagrange multipliers. By ©, 2x = -1 (24+y) g 2y = -f(x+2y) Toking ratio of these two equations, x-2xty (Assuming ito) y atry (If dao, by to x=y=0 which contradicts 0 ).

P. 2 ::(+2y) = y(x+y) x trung= 2ang ty? v = ... = . - Piut y = x in D, + x +7' _4=; 7:3x? -4 = 0 -34-4 are are critical points - Put y=-xin x txt=0 x = 12 .: (2,-), (2, 2) are critical points off Also, f(tim, tos)* + +2=te =ht & f (2,-2) = f(2)= 4 + 4 +2=10 : Masinan value of f is so of minionpon value is 14.