Visualizing Lagrange multipliers Sorry if this seems like a very basic question but I am having trouble visualizing Lagrange multipliers. Particularly the equation:
$ \nabla f = \lambda * \nabla g $
f = function to maximise. g = constraint.
I don't understand why equating the gradients in such a way produces the extremum. I watched a Khan Academy video and the explanation was as follows:
Plotting Contours of f and g
My question is:
Why does the extremum occur only where the contours of f touch the constraint at one point? Why can they not occur where there are multiple points? For example:
Ex: Contours of f meeting g at two points
Also, why is it that equating the gradients in such a way produces the point where they touch at only one point? My understanding is $ \nabla $f is the gradient vector of f and $ \nabla $g is the gradient vector for g. It seems to be there may be infinitely many points which may satisfy the equation but are not necessarily the extremum. 
Kindly help me visualise what is going on here. Thanks in advance.
EDIT:
My understanding of gradients

I was thrown off by this Medium article:
https://medium.com/@andrew.chamberlain/a-simple-explanation-of-why-lagrange-multipliers-works-253e2cdcbf74
I had been incorrectly visualising the gradient of a 2D function in 3D.
Although most of the content is correct, the diagram is slightly misleading in the representation of the gradients. Had to revisit the definition of gradients which I did through this post: What is a Gradient?. Finally, by piecing everything together what I see is: for a function f(x1, x2) the gradient vectors must be "projected" onto the x1-x2 plane. On that plane, if $ \nabla f $ = $\lambda \nabla g $ (The vector are colinear) then we've reached a stationary point which may yield an extremum.
