Minimizing a simple non-linear function by using Lagrange Multipliers

Given Function:

`5 − ( x − 2 )^2 − 2 ( y − 1 )^2`

Constraints

`x + 4 y = 3`

Solution:

According to Lagrange Theory,

`L = f ( x ) + λ g ( x )`

f(x) is the given function

g(x) is the given constraint

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

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

To successfully minimize the given function, we have to partially derivate L with respect to x, y and `λ`

`(∂ L)/ (∂ x) = 0`

`− 2 x + 4 + λ = 0`.........................................................(equation 1)

`(∂ L)/ (∂ y) = 0`

`− 4 y + 4 + 4 λ = 0`

`− y + 1 + λ = 0`...........................................................(equation 2)

`(∂ L)/( ∂ λ) = 0`

`x + 4 y = 3`..................................................................(equation 3)

`Equation (1) - Equation (2)`

`− 2 x + 4 + λ + y − 1 − λ = 0`

`− 2 x + y + 3 = 0`

`y = 2 x − 3`

substituting y in equation 3

`x + 4 y − 3 = 0`

`x + 8 x − 12 − 3 = 0`

`x = 5/3`

substituting x

`y = 2 x − 3`

`y = 2* ( 5/ 3 ) + 3`

`y = 1/3`

substituting x & y

`− 2 x + 4 + λ = 0`

`− 10/ 3 + 4 + λ = 0`

`3 λ = − 10 + 12`

`λ = 2/ 3`