I've been trying to understand the principles behind the Lagrangian multipliers and I think I've got a rough understanding of it. Would appreciate it if you guys could help me answer a few questions!
I've pretty much self-studied this from here and here.
As far as I understand it, the Lagrangian multiplier essentially works by ensuring the gradient of the function is equal to the gradient of the restraint. Assuming that $g(x,y) = c$ is the restraint, it also ensures the point satisfies the restraint.
However, I don't understand the reasoning behind some of the statements in the ideashop link:
1) "The most important thing to know about gradients is that they always point in the direction of a function's steepest slope at a given point." . Apparently a gradient is a collection of partial first derivatives but I don't understand how one gets from a collection of partial first derivatives to 'Always pointing in the direction of the steepest slope'. Wouldn't it be changing direction as one went around a 3D surface?
2) Why is the gradient always perpendicular to the level curve?
3) How does one get from $$\nabla L=\begin{bmatrix}\frac{\partial f}{\partial x_1}-\lambda\frac{\partial g}{\partial x_1}\\\frac{\partial f}{\partial x_2}-\lambda\frac{\partial g}{\partial x_2}\\g(x_1,x_2)-c\end{bmatrix}=0$$ to $$L(x_1,x_2,\lambda)=f(x_1,x_2)-\lambda(g(x_1,x_2)-c)$$ (Both sourced from ideashop) ?
Thank you for your help in advance!