LaGrange Multiplier Derivative vs Function

Question

Why do LaGrange Multipliers produce functional extrema when they look for derivative extrema? I see the process of setting constraint and function gradient vectors equal with lambda as a scaling factor and using the original constraint to solve the system. Doesn't this just yield points of highest gradient and not necessarily points of extreme value for the original function? I can't see any other questions on this so I feel I may be wildly misinterpreting something.