罗尔定理应用中辅助函数的构造

The construction of auxiliary functions using Rolle′s theorem

应用罗尔定理证明问题时常通过构造一个与问题相关的辅助函数达到解决问题的目的,这既是证明的需要,也是证明的关键.构造辅助函数没有固定的模式与方法,具有较强的技巧性,是高等数学学习的一个重点内容,也是一个难点问题.给出凑原函数法、乘积因子法、不定积分法、微分方程法、常数k值法等5种常用的罗尔定理应用中构造辅助函数的方法,使辅助函数的构造有章可循,为教学提供参考.

When applying Rolle＇s theorem to prove mathematical problems, constructing a problem-related auxiliary function is one of the most common ways, which is not only necessary but critical for the mathematical proof. However, the construction of auxiliary functions usually doesn＇t have fixed patterns and methods but needs skillful techniques. It is a key content and difficult point during the process of learning and teaching advanced mathematics. Discussed five common used methods for constructing the auxiliary function when applying Rolle＇s theorem, including primitive function method, multiplication factor method, indefinite integration method, differential equation method and constant k-value method. With the help of in-depth understanding of those methods, proposed some guiding principles for the construction of auxiliary functions when using Rolle＇s theorem, which provide insights for teaching and learning advanced mathematics in practice.