Is the Lagrange multiplier $\lambda$ always positive? In the Lagrange function
$${\mathcal {L}}(x,y,\lambda )=f(x,y)-\lambda g(x,y),$$
is the Lagrange multiplier $\lambda$ term supposed to always be positive or can't it take negative values?

The sign (and value) of $\lambda$ signifies what happens to the value of the extremum you find when you change the constraint by a tiny amount, say from $g(x,y)=0$ to $g(x,y)=\epsilon$. That's it. Positive or negative, large or small doesn't matter. If you have found a $\lambda$ (i.e. if you have found an $x$ and a $y$ for which a $\lambda$ exists), you have found an extremum.
