Analysis of the problem solving problem of inequalities in abstract functions
抽象函数中不等式竞赛题解法例析

摘要

抽象函数属于数学竞赛里的高频考查知识点。抽象函数不等式类问题，因自身具备的抽象性、灵活性、技巧性、综合性与隐蔽性等特征，成了能有效检验学生逻辑推理能力和函数性质应用能力的重要题型，针对抽象函数不等式竞赛题的常见题型，可对六种核心解题方法做系统的梳理与分析，这六种方法分别是单调性法、赋值法、构造法、柯西法、迭代法以及凹凸性法。本文选取全国高中数学联赛、国内外各类经典竞赛真题和改编习题作为具体案例，结合每一道例题展开细致的解题解析，能为参与数学竞赛的学生梳理清晰的解题思路、提供实用的方法指导，也能让学生更从容地应对各类抽象函数不等式竞赛题目。其中单调性法与构造法适用范围最广，柯西法与迭代法对特定函数方程结构更具针对性，合理选择与组合这些方法可显著提升竞赛解题效率。

Abstract

Abstract functions belong to the high-frequency examination of knowledge points in mathematics competitions. Abstract function inequality problems, due to their own abstraction, flexibility, skill, comprehensiveness and concealment, have become an important question type that can effectively test students' logical reasoning ability and function property application ability. This paper selects the National High School Mathematics League, various classic competition questions and adapted exercises at home and abroad as specific cases, and carries out detailed problem-solving analysis in combination with each example problem, which can sort out clear problem-solving ideas for students participating in mathematics competitions, provide practical method guidance, and allow students to deal with various abstract function inequality competition questions more calmly. Among these methods, the monotonicity method and the construction method have the broadest applicability, while the Cauchy method and the iteration method are more targeted towards specific types of functional equations. Properly selecting and combining these methods can significantly improve problem-solving efficiency in competitions.