Constraint Minimization using Lagrange Multipliers

To Minimize the fuction 5-(x-2)^2-2(y-1)^2 subjected to constraint x+4y = 3 using Lagrange multiplier method.

Solution: To solve this problem we will use Lagrange method for constrained minimization

According to Lagrange,

L = f - λg

Where,

f = 5 - (x-2)^2-2(y-1)^2 and g = x+4y = 3

Therefore,

L = 5 - (x-2)^2 - 2 (y-1)^2 - λ(x+4y-3)

L = 5 - (x-2)^2 - 2 (y-1)^2 - λx - λ4y + λ3 .............(1)

To solve this we need to find the partial derivates of \"L\" with respect to x, y and λ.

Let,

∂L/∂x = ƒ(x), ∂L/∂y = ƒ(y), ∂L/∂λ = ƒ(λ) , now from equation (1)

step -1 : ƒ(x) = - 2(x-2) - λ = 0

= -2x + 4 - λ = 0 ; By solving this x = -(λ - 4)/2 ............(2)

step-2 : ƒ(y) = - 4(y - 1) - 4λ = 0

= -4y + 4 - 4λ = 0 ; By solving this y = -(λ - 1) ...............(3)

step -3 : ƒ(λ) = -(x +4y-3) = 0 ............ (4) ; Subsitute the values of x and y in

equation (4);

x+4y-3=0

-(λ-4)/2+4(-λ+1)-3=0

-λ/2-4λ=-3

-9λ/2=-3

-9λ=-6

λ=2/3

Now if we subsitute the value λ = 2/3 in equation (2) and (3),

x = 5/3 and y = 1/3

therefore

for x = 5/3, y = 1/3, λ = 2/3 the function 5 - (x-2)^2- 2 (y-1)^2 will be minimized.