## Constraint Minimization of Equation using lagrange's method

OBJECTIVE

The main objective is to Minimize the given equation with respect to the given constraint.

Problem:

This function f(x,y) = 5-(x-2)^2 -2(y-1)^2 is subject to the following constraint x + 4y = 3

Solution:

Make a new function as 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)

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

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

Perform partial differential equation on this g(x,y) function w.r.t. x, y & lamda

(del g)/(del x) = 0

-2x + 4+ lamda = 0   ---------------> Eq. 1

(del g)/(del y) = 0

-4y + 4+ 4lamda =0

lamda - y+1 =0

lamda = y-1              ---------------> Eq. 2

(del g)/(del lamda) = 0

x+4y-3=0                ---------------> Eq. 3

Put eq. 2 into eq. 1, we get

-2x+4+y-1=0

-2x+y+3=0              ---------------> Eq. 4

multiply eq. 3 with 2 and solve with eq. 4, we get

9y-3=0

y=1/3

put value of y in eq. 2, we get lamda

lamda = -2/3

put value of y in eq. 3, we get  value of x

x = 5/3

CONLUSION

We get the value of x , y and lamda

x = 5/3 , y=1/3 , lamda = -2/3

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