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

Suppose $x_0$ is an extremum, then the constraint contour $g(x) = 0$ and cost contour $f(x)=f(x_0)$ pass through the point $x_0$.
If the contours are not 'parallel' at $x_0$, that is suppose they cross. Then you can see that there are points on the contour $g(x) = 0$ that have higher and lower values of $f$,
which contradicts $x_0$ being an extremum.
So, the contours must be 'parallel' at $x_0$ which means the gradients must be collinear at $x_0$.
