Here is a picture of what the level curves of f and the curve defined by g(x,y)=c might look like if the levels of f and the curve g=c are not tangent at the point we are interested in:
In this picture, the level of f running through (2,1) intersects the curve g=c with some non-zero angle. This means that nearby levels of f will intersect the curve also; so we can increase the value of f by moving along g in the direction of the increasing levels, and decrease the value of f by moving in the opposite direction. This means f can have neither a max or a min at (2,1).
So if f has a max or min at some point of g=c, the level of f must be tangent there.
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