This is the most important and useful method for optimization.it can be used to optimize function that depends on a number of independent variable and when functional constraints are involved. As such, it can be applied to wide range of practical circumstances provide the objective function and the constraints can be expressed as continuous and differentiable functions. 

 

   The method of Lagrange multiplier basically converts the preceding problem of finding the minimum or maximum into the solution of a system of algebraic equations,  thus providing a convenient scheme  to determine the optimum. 

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Here is the pdf that explain Lagrange multiplier: