Is the Lagrangean multiplier always finite?

Question

In the constrained maximization problem $$\mathcal{L}(x,y) = f(x,y) + \lambda \cdot (M - xp_x - yp_y)$$ where $x$ and $y$ are goods with prices $p_x$ and $p_y$, can $\lambda = +\infty$? My conjecture is that $\lambda$ is finite iff there is an interior solution and infinity otherwise, but I cannot find a reference.

Answer 1

The idea behind Lagrange multipliers is that at an extremal point of a function along a constraint, the gradient of the function is orthogonal to the constraint. The method does this by constructing a normal vector to the constraint, and checking to see if there is any point on the constraint where the gradient is some $\lambda\in\Bbb R$ times the normal vector.

Since the gradient is finite (where it exists), and the normal vector is non-zero, we must have $\lambda$ finite, if it exists.