Why is the function either maximum or minimum when the constraint is tangential to the function contour?

Question

Given a function $f(x,y)$ and a constraint $g(x,y)=c$, Assuming the fact that the function either maximum or minimum when the constraint is tangential to the function contour, I get why the gradients are proportional, namely $\nabla f = \lambda \nabla g$ . But I don't understand in the first place why the function attains its critical points on the constraint only when they are tangential.

Answer 1

From Wikipedia:

"The method of Lagrange multipliers relies on the intuition that at a maximum, $f(x,y)$ cannot be increasing in the direction of any such neighboring point that also has $g(x,y)=c$. If it were, we could walk along $g(x,y)=c$ to get higher, meaning that the starting point wasn't actually the maximum. Viewed in this way, it is an exact analogue to testing if the derivative of an unconstrained function is $0$, that is, we are verifying that the directional derivative is $0$ in any relevant (viable) direction."

Note that the equations of the Lagrange Multiplier method are necessary but not sufficient conditions for an optimum (on a boundaryless domain). If they have multiple solutions, you will have to compare the solutions further and/or examine their Hessians.