拉格朗日算子的直观解释

n the following paragraph, we discuss the method of Lagrange multipliers( 拉格朗日乘子法 ), for solving the problem of conditional extreme values. 

Assume the object function to be  and the constraint ( 约束条件 ) to be  . Because represents a plane curve, we may replace  by expression  
In this case, by substitution, we have 

Therefore    

  

The necessary condition for    having extreme value is  , that is 

This means that if  has extreme values at  , the following two equations must simultaneously hold in  and  . 

This system in fact means that the tangent vector    is at the same time perpendicular to vectors    and     And  . Thus there exists a parameter   such that 
or     . Hence we finally have 

This system is the necessary condition for the function    having extreme values at