Constrained Minimization Problem - Pen and Paper Calculation

AIM of the project:

To minimize the following function using Lagrange multipliers.

f(x,y)=5-(x-2)^2-2(y-1)^2

Constraint:x+4y=3

The new function with Lagrange multiplier is as follows:

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)

Now, we must calculate the differential of the above function with respect to x, y and and equate

them to zero, since at minima the differential of a function is zero.

dg/dx= -2(x-2)+ λ=0 => λ-2x+4=0…………………………………………………1

dg/dy= -4(y-1)+4λ= 0 = > λ-y+1=0………………………………………………….2

dg/dλ=x+4y-3= 0 =>x+4y=3……………………………………………………….3

Solving the first two equations,

We get, 2x-y=3…………………………………………………4

Solving the equations 3 and 4,

We get x = 5/3 and y =1/3.

The value of the function at (5/3,1/3) is 4.

The constraint also gets satisfied with these values.

5/3+4*1/3=9/3=3