拉格朗日乘数法作为一种解决条件极值问题的优化算法，在实际问题中占有重要地位，各类数学分析教程中都介绍了该方法。尽管学生可以很快掌握它，但绝大多数人对其始末缘由处于一知半解的状态。本文站在初学者的角度，首先从极值的必要条件入手，利用微分法，由浅入深地探讨条件极值的必要条件，进而引出拉格朗日乘数法；其次，结合极值点的充分条件，利用二阶微分的正负号判断该点是否为极值点；最后，介绍了拉格朗日乘数法在一些典型问题中的应用。旨在帮助学生透彻地理解和掌握该方法。因其具有较强的实用性，故它有利于培养学生的发散性思维和创新性思维，能有效提高学生分析问题、解决问题的能力。
As an optimization algorithm to solve conditional extremum problem, Lagrange multiplier method plays an important role in practical problems. This method is introduced in various mathematical analysis tutorials. Although learners can quickly grasp the method, most people are in a state of half-understanding of its cause and effect. From a beginner’s point of view, firstly this paper discusses the necessary conditions of conditional extremum problem using differential method, based on the necessary conditions of extremum problem. Then, the Lagrange multiplier method is discussed. Secondly, combined with the sufficient conditions of extremum point, the positive or negative signs of the second-order differential are used to judge whether the point is an extremum point. Finally, the application of Lagrange multiplier method in some typical problems is presented. The purpose is to help learners understand and master the method thoroughly. This method has strong practicability, so it is helpful to cultivate students’ divergent thinking and innovative thinking, and can effectively improve students’ ability to analyze and solve problems.
