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:
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:
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.