Alternative to Lagrange multipliers

Question

We want to maximise $f(x,y)$ under the constraint $g(x,y)= 0$. Instead of using Lagrange multipliers, can we just set the dot product of the $\nabla f \cdot{ (1,dy/dx)}$ under the constraint $g$ to equal $0$. This should work because the vector $(1,dy/dx)$ is parallel to the contour $g(x,y) =0$ and perpendicular to the $\nabla{ f}$.

Answer 1

This mostly works, but what if ${dy\over dx}$ doesn’t exist at some point on the constraint curve? This isn’t all that far out a possibility: consider the points at which $x^2+y^2=1$ intersects the $x$-axis, for instance.

You either need to consider these cases separately, or use a slightly different formulation if you wish to avoid introducing a Lagrange multiplier: if $\nabla f$ and $\nabla g$ are parallel, then $$\begin{vmatrix}{\partial f\over\partial x} & {\partial f\over\partial y} \\ {\partial g\over\partial x} & {\partial g\over\partial y}\end{vmatrix} = 0.$$ That is, use a cross product instead of a dot product.