Constraint Minimization - Pen and Paper calculation

Aim:

To minimize the following function:

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

Subject to 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) + lamda(x + 4y - 3)`

`g(x,y) = 5-(x-2)^2 -2(y-1)^2 + lamda(x+4y-3)`

The above is the new function which is to be minimized.

The 3 new equations:

=> `delg//delx = 0` ; `delg//dely = 0`; `del g//del lamda = 0`

=> `delg//delx : -2x + 4 + lamda = 0` ------- equation 1

=> `delg//dely: -4y + 4 + 4lamda = 0` -------- equation 2

=> `delg//dellamda = x + 4y -3 = 0` --------- equation 3

equation 2 - `4*`equation 1

=> `-4y + 4 + 4lamda - (-8x + 16 + 4 lamda)`

=> `-4y +4+4lamda+8x-16-4lamda`

=> `8x - 4y -12`

=> `2x-y-3 = 0` ------ equation 4

from equation 4 :

=> `2x-y=3`

=> `2x = y + 3`

=>`x = (y + 3)//2`

putting the above value of x in equation 3

=> `y + 3//2 + 4y = 3`

=> `y + 3 + 8y = 6`

=> `9y = 3`

=> ``y = 1//3``

to get the value for x putting the value of y in equation 3:

=> `x + 4/3 = 3`

=> `3x + 4 = 9`

=> ``x = 5/3``

therefore `lamda = -2/3`