Question

Minimize the function f(x,y)=x²+y²-xy subject to the constraint x+2y-14=0

(we are studying the Lagrangian Multiplier if that helps!) Minimize the function f(x,y)=x²+y²-xy subject to the constraint x+2y-14=0

(we are studying the Lagrangian Multiplier if that helps!) @Mathematics

Answer 1

add in an L then

Answer 2

f(x,y,L) =x²+y²-xy+L(x+2y-14)

and take all your partial derivatives set to 0

Answer 3

fx = 2x -y+L = 0

fy = 2y - x + 2L = 0

fL = x+2y-14 = 0 but replace x and y by terms of L

Answer 4

what do you mean by replace x and y by terms of L?

Answer 5

fx = 2x -y+L = 0
2x = y-L
x = (y-L)/2

fy = 2y - x + 2L = 0
x = 2y+2L

2y+2L = (y-L)/2

4y +4L = y-L

4y-y = -L-4L

3y = -5L

y = -5L/3

x = 2(-5L/3)+2L

x = -10L/3 +2L

x = -10L+6L/3

x = -4L/3

Answer 6

fL = x+2y-14 = 0

fL = (-4L/3)+2(-5L/3)-14 = 0

fL = -4L/3 -10L/3 -14 = 0

fL = -14L/3 -14 = 0

fL = -14L/3 = 14

fL = -14L = 42

fL = L = 42/-14 = -3

Answer 7

any of that make sense?

Answer 8

yes okay so I sub x and y into the parrtial derivative of L?

Answer 9

essentially you solve a system of equations between fx and fy so that they are in terms of L, your lagrange multiplier

then use those in your constraint to determine the value for your L

Answer 10

then back substitute it to determine x and y

Answer 11

yes

Answer 12

if i did it right, x = 4 and y = 5; otherwise i missed a sign and they might need to go negative