A substitution-based perspective: Solving constrained extremum problems through variable transformation
“换”然一新：换元法解条件最值问题

摘要

换元法是中学数学中用以简化问题结构、转化未知形式的核心思想方法。本文以2022年全国新高考Ⅱ卷第12题为研究载体，详细阐述了目标换元、和差换元、三角换元、比值换元、均值换元等多种换元技巧在求解二元二次约束条件下最值问题中的具体应用，完整推导出本题的结论，选项B、D正确。通过一题多解的方式，对比各类换元法适用条件：目标换元求解线性目标，三角换元适配可配方平方和约束，和差换元简化对称方程，比值换元处理齐次条件，均值换元依赖恒等变形降元，揭示了换元法“化繁为简、聚焦本质”的核心价值；同时进一步追溯题源、引申推广，并结合教学实践提出了相应的思考，为高中数学教学提供参考与借鉴。

Abstract

The substitution method serves as a core ideological approach in middle school mathematics to simplify problem structures and transform unknown forms. Taking the 12th question of the 2022 National New College Entrance Examination Volume II as the research carrier, this paper elaborates on the specific applications of various substitution techniques—including target substitution, sum-difference substitution, trigonometric substitution, ratio substitution, and mean value substitution—in solving maximum and minimum value problems under binary quadratic constraint conditions, and fully deduces the conclusions of the question, verifying that Options B and D are correct. Through multi-solution analysis of a single problem, this paper compares the applicable conditions of different substitution methods: target substitution is suitable for solving linear objectives, trigonometric substitution adapts to constrained conditions of configurable sum of squares, sum-difference substitution simplifies symmetric equations, ratio substitution addresses homogeneous conditions, and mean value substitution realizes dimension reduction via identical deformation. It reveals the core value of the substitution method in “simplifying complexity and focusing on essence”. Furthermore, this paper traces the problem source, extends and generalizes the relevant conclusions, and puts forward corresponding reflections combined with teaching practice, providing references for senior high school mathematics teaching.