二元函数极值临界问题的分层递进式教学设计

A Progressive Teaching Design for Critical Problems of Extreme Values in Binary Functions

摘要: 针对多元函数极值教学中，学生面临的概念理解模糊、方法应用机械及临界情形处理无序的三重困境，本文提出“分层递进式”教学设计。该设计将临界问题(即二阶充分条件失效时)的系统化处理作为突破口，构建了“定义法”、“降维转化法”与“高阶导数法”三个教学层次。第一层次回归极值定义，通过邻域比较强化直观认知；第二层次运用坐标变换，将二元问题降维为一元问题，渗透转化与化归思想；第三层次引入泰勒展开，构建高阶导数判别工具。三个层次相互关联、逐层递进，旨在突破传统教学中“遇到即放弃”的思维定势，完善极值判定的知识体系，并在此过程中培养学生综合运用数学思想方法解决疑难问题的能力，为创新人才培养提供可操作的教学路径。

Abstract: In the teaching of extreme values of multivariate functions, students frequently confront three interrelated predicaments: ambiguous conceptual comprehension, mechanical application of methods, and unstructured handling of critical cases (i.e., scenarios where the second-order sufficient condition fails). To address these challenges, this paper proposes a “hierarchical progressive” instructional design, which takes the systematic resolution of critical cases as a breakthrough point and constructs three interlinked teaching layers: the “definition-based approach”, “dimension-reduction transformation method”, and “higher-order derivative method”. The first layer revisits the definition of extreme values, reinforcing intuitive understanding through neighborhood comparison. The second layer employs coordinate transformation to reduce bivariate problems to univariate ones, embedding the mathematical ideas of transformation and reduction. The third layer introduces Taylor expansion to develop a discriminant tool based on higher-order derivatives. These three layers are logically connected and progress incrementally, aiming to break the traditional teaching mindset of “abandoning when encountering difficulties”, improve the knowledge system for extreme value determination, and cultivate students’ ability to comprehensively apply mathematical thinking and methods to solve complex problems. This design provides an actionable teaching pathway for the cultivation of innovative talents.