I have read in several places that the sign of the Lagrange Multiplier is unimportant for finding the critical points of the Lagrangian. However, I don't believe that this is entirely true. In my professor's lecture, we were presented with the following setup:
Maximize $f(x)$ subject to the constraint $g(x)\geq0$, where the Lagrangian is expressed as $L(x, \lambda)=f(x)+\lambda g(x)$, $\lambda g(x)\geq0$, and $\lambda\geq0$.
Supposing for simplicity that $f(x)$ and $g(x)$ are each both concave and differentiable, then the value $x_0$ solves the max problem iff $x_0$ satisfies the first-order condition of the Lagrangian. However, to solve the first-order condition of the Lagrangian, it would have to be true that $-f'(x_0)=\lambda g'(x_0)$. But $\lambda$ is non-negative! Is it then true that $f'(x_0)$ and $g'(x_0)$ cannot have the same sign?