Question about calculus of variation.

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What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?

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Not sure what you mean. The difference is one is the maxima and one the minima. Perhaps you mean in the method? Then there is no difference initially. It is similar to finding the local maxima and minima for a differentiable function of one variable. You find the extreme points in the first pass, then you need other methods to determine what those extreme points are (local max or min).