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?

Geometrically, the Lagrange multiplier method has the following interpretation:


*

*We want a point on the level curve $g(x,y)=0$. This comes from differentiating with respect to $\lambda$ and setting to zero.

*We want that, at the point $p$ under consideration, the level curve of $f(x,y)$ is tangent to the level curve $g(x,y)=0$. This is equivalent to the normal vectors to the level curves, given by the gradients, are parallel. In other words, we require the existence of a scalar $\lambda$ (necessarily different from zero if we assume a minimum of regularity for the level curves) such that
$$\nabla_{p} f = \lambda\nabla_{p} g$$
From condition $2$, it is clear that there is no restriction telling us that $\lambda$ must e positive or negative. On the contrary, with this intuition it is easy to construct examples for both cases.
