Constraint Minimization

AIM:

To minimize the following Function:

f(x,y)=5-(x-2)^2-2〖(y-1)〗^2

CONSTRAINT:

x+4y=3

f(x,y)=5-(x-2)^2-2〖(y-1)〗^2

5-(x^2+4-4x)-2(y+1-2y)=0

5-x^2-4+4x-2y^2-2+4y=0

-x^2-2y^2+4x+4y-1=0

By Lagrangian Method:

g(x,y)=f(x,y)+λ(x+4y-3)

g(x,y)=5-(x-2)^2-2(y-1)^2+λ(x+4y-3)

Differentiating the equations:

∂g/∂x=0

∂g/∂y=0

∂g/∂λ=0

∂g/∂x:-2x+4+λ=0 -EQUATION 1

∂g/∂y:-4Y+4+4λ=0 -EQUATION 2

∂g/∂λ:x+4y-3=0 -EQUATION 3

EQUATION 2 - 4* EQUATION 1

-4Y+4+4λ-(-8X+16+4λ)

8X-4Y-12

2X-Y-3=0

2X=Y+3

X=(Y+3)/2

Putting The value of X in EQUATION 3

(Y+3)/2+4Y=3

Y+3+8Y=6

9Y=3

Y=1/3

To Get the value of X

2X=10/3

X=5/3

SUBSTUTING THE VALUE OF Y IN EQUATION 3

λ=-2/3

The values of

X= 5/3
Y=1/3
λ=-2/3