I'm trying to understand why the method of Lagrange multipliers is legit.
If I'm trying to find min/max of a function $f(x,y)$ subject to constraint $g(x,y)=c$
then as per my professor $\mathbf {at \ the \ constrained \ min/max, \ the \ rate \ of \ change \ of \ f \ must \ be \ zero \ in \ any \ direction \ along \ g=c}$.
I cannot seem to understand this statement.
But this is the elementary part of the proof;
coz based on this he says $\frac{df}{ds}_\hat u = \nabla f.\hat u = 0$ where $\hat u$ is tangent to g=c
$\mathbf {Why \ is \ \hat u\ tangent \ to \ g=c ? \ Is \ this \ based \ on \ the \ fact \ that \ at \ the \ constrained \ min/max \ g=c \ is \ tangent \ to \ f?}$
and the rest is understable that such $\nabla g$ is perpendicular to $\hat u$ that is the tangent and hence $\nabla f$ and $\nabla g$ are parallel finally yielding that $\nabla f = \lambda \cdot\nabla g $
