Analysis of examples in solving high school function problems using “constructing functions and comparing sizes”
“构造函数，比较大小”之解高中函数题例析

摘要

本文以“构造函数”为核心，系统探讨其在解决高中函数比较大小问题中的应用。通过分类归纳与例题分析，重点阐述两种基本题型：作差构造函数求参数范围、作差构造函数证明不等式。研究发现，将复杂双函数不等关系转化为对单一构造函数单调性、极值与最值的分析，是问题转化的关键。该方法提供通用框架，揭示“转化与归约”思想本质，即把陌生综合问题化为能用导数研究的熟悉模型。本文结合例题拆解构造函数的思维起点与逻辑链条，旨在助学生理解方法来源，提升其分析、转化问题的数学素养。

Abstract

This paper focuses on “constructing functions” and systematically explores their application in solving high school function comparison problems. Through classification, summarization, and example analysis, it emphasizes two basic problem types: using the difference function to find the parameter range and using the difference function to prove inequalities. The study finds that transforming complex inequalities between two functions into the analysis of the monotonicity, extrema, and maximum/minimum values of a single function constructor is key to problem transformation. This method provides a general framework, revealing the essence of the "transformation and reduction" idea—that is, turning unfamiliar comprehensive problems into familiar models that can be studied using derivatives. This paper breaks down the starting point and logical chain of constructing functions using examples, aiming to help students understand the source of the method and improve their mathematical literacy in analyzing and transforming problems.