Constraint Minimization - Pen and Paper calculation

Aim:

To Minimize the following function

5-(x-2)^2 -2(y-1)^2

subject to the following constraint

x + 4y = 3

The method of Lagrange multipliers is a method for finding max or min of a function of several variables restricted to a given subset.

Method of Lagrange Multipliers:

Suppose that f(x, y) and g(x, y) is differentiable. To find the local maximum and minimum values of f subjected to the constraint g(x, y) = 0, find the values of x. y and λ that simultaneously satisfies the equations.

`Δf = λ Δg` and `g(x, y)= 0`

Constraint `x + 4*y = 3`

So we have `g(x, y)=x + 4*y -3 = 0`

`L= f(x,y)- λg(x,y)`

Where λ is called the Lagrange multiplier.

Solving for the critical points

`Lx =(∂L)/(∂x) = -2(x-2)-λ = 0`

`x = 2-(λ/2)`

`L_y =(∂L)/(∂y) = -4(y-1)-4λ=0`

`y = 1- λ`

substituting x and y values in g(x, y) constraint equation

`x + 4*y -3 = 0`

`2-( λ/2) + 4(1- λ) - 3 = 0`

`4λ + (λ/2) = 3`

`λ =2/3`

now find the x and y values using λ:

`x = 2 - (λ/2)`

`x = 2 -(1/3)`

`x = 5/3`

`y = 1- λ`

`y = 1-2/3`

`y =1/3`

function f(x, y) minimum value in the constraint g(x, y) is

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

`f(5/3,1/3)=5-(5/3-2)^2-2(1/3-1)^2`

`f(5/3,1/3)=4`

The function has a minimum value of 4 in the constraint of g(x, y).