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I've always used the method of Lagrange multipliers with blind confidence that it will give the correct results when optimizing problems with constraints. But I would like to know if anyone can provide or recommend a derivation of the method at physics undergraduate level that can highlight its limitations, if any.

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    @John, you may or may not find the answers to this similar question helpful: http://math.stackexchange.com/q/674/4002011-02-27

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