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Abstract:In this paper we summary mainly several kind of methods about conditional extremum of function of several variables,this topic is the important constituent part;and discuss the applications of some problems about the proof of inequality,physics,production and market etc..
In our paper we study mainly several kind of methods about conditional extremum of function of several variables,such as direct substitution method,Lagrange multiplier method,Cauchy inequality,gradient method,numbers and figures  combined method etc.;and introduced mainly total differentiation and second partial derivative,i.e. Hesse matrix method etc. about Lagrange multiplier method and in order to reference and use for reference in solving corresponding problems,and find  the methods of solving problems faster and more precise.
Key words:Lagrange multiplier method,Cauchy inequality,extremum,applications.

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