Constrained Minimization Problem

Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints. One of the most common problems in calculus is that of finding maxima or minima (in general, "extrema") of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside conditions or constraints. The method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.

Minimize F(x) = `5-(x-2)^2 -2(y-1)^2`

= `-x^2-2y^2 + 4x+4y-1`

Constraint = `x+4y =3`

Using Lagrange Approach to Minimize this function:

`L(x,y,lambda)`= `F(x) + lambda(x+4y-3)`

`L(x,y,lambda)`= (`-x^2-2y^2 + 4x+4y-1` ) + `lambda(x+4y-3)`

`(dell)/(delx) = -2x +4 +lambda = 0`------------------------------------------------------Equation 1

`(dell)/(dely) = -4y +4 +4lambda = 0`-----------------------------------------------------Equation 2

`(dell)/(dellambda) = x+4y-3=0`------------------------------------------------------------Equation 3

Solving Equation 1 ,2 ,3 we get the value of x,y,`lambda`:

`x = 5/3`

`y = 1/3`

`lambda = -2/3`

For these value the function F(x) is minimum