Constraint Minimization

AIM: To minimize the given function subject to the given constraint.

FUNCTION `=5-(x-2)^2-2(y-1)^2`

Subject to constraint:- `λ (x + 4y-3)`

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

And `g(x,y)=f(x,y)+λ (x + 4y-3)`

Therefore,

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

So, Differentiating the above equation w.r.t `x,y and λ` respectively for minimizing the function.

`(∂g)/(∂x)=0 ∶ -2(x-2)+ λ=0`

Or, `-2x + 4+ λ=0……………….eq(1)`

`(∂g)/(∂y)=0 ∶ -4(y-1)+ 4λ=0`

Or, `-4y+4+4 λ=0`

`-y+1+λ=0……………….eq(2)`

and

`(∂g)/(∂ λ)=0 ∶ x + 4y=3……………….eq(3)`

Now by doing `eq(1)-eq(2)`

We get, `-2x + 4+ λ- (-y+1+λ) =0`

Or `-2x+y+3=0`

So, now by solving `eq(3) and eq(4)`

We get `x= 5/3 ,y= 1/3`

and we can calculate the value of `λ` by putting values of x in eq(1) and y ineq(2).

The value of `λ= -2/3`

And by putting these values in `eq(3)` , we get

`(5/3) + 4(1/3)=3`

`3=3`

which satisfies the equation and our function is also minimized.