Use Lagrange multipliers to find the minimum value of the function f(x,y)=x^2+y^2 subject to the constraint xy=3

Answer 1

f(x,y)=x²+y² g(x,y)=xy=3

First, we impose

∂f_x=λ∂g_x ∂f_y=λ∂g_y

Then we solve these equations

∂f_x=λ∂g_x → 2x/y=λ

∂f_y=λ∂g_y →2y/x=λ

therefore x/y=y/x →x=y

Now we can eliminate y by replacing it with x since they are equal

f(x,y)=2x² g(x,y)=x²=3

and so x=√3 and f_min=6

Note: we would get the same answer by eliminating x instead