Constraint Minimization - Pen and Paper calculation

Problem statement:

Minimize the following function

5-(x-2)^2 -2(y-1)^2

subject to the following constraint

x + 4y = 3

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

Which can be written by

`f(x,y)=-x^2-2y^2+4x+4y-1`

Now,

`f(x,y)=-x^2-2y^2+4x+4y-1`

λ is the Lagrange operator.

Now,

`∂g/∂x= -2x+4+λ=0` - eq1

`∂g/∂y=-4y+4λ+4=0` -eq2

`∂g/∂λ=x+4y-3=0` -eq3

Solving above eq1 and 2

We get,

`x=λ/2+2` -eq4

`y= λ+1` -eq5

substituting values of x,y in equation 3,

`y= λ+1`

we get

`λ= -2/3`

substituting value of λ in eq.4 & eq.5

`x=5/3 and y=1/3`

RESULT: so, at `x=5/3 and y=1/3` the `f(x,y)` is minimized and also satisfies the constrain equation `x+4y=3`.