双勾函数在求解最值问题的探索研究

Exploration and Research on Double Hook Function in Solving Extremum Problems

摘要: 随着数学学习的发展与深入，形如$f\left(x\right)=x+\frac{a}{x}\left(a>0\right)$ 的模型已经多次出现在我们的视野中，初等数学中求最值问题逐渐普遍化且考查力度较大，刻板印象就是直接用基本不等式求解，因为这通常是同学们最希望看到的容易求解形式。事实上，这类式子是一类双勾函数，本文将剖析双勾函数的基本性质，并利用这些性质解决基本不等式无法解决的最值问题。

Abstract: As our study of mathematics progresses, the model represented by $f\left(x\right)=x+\frac{a}{x}\left(a>0\right)$ has repeatedly come into our field of vision. In elementary mathematics, the problem of finding the maximum or minimum value has become more common and more challenging. The stereotype is to directly use the basic inequality to solve it, as this is often the easiest form that students hope to see. In fact, such expressions are a class of double-hook functions. This article will analyze the basic properties of double-hook functions and use these properties to solve maximum or minimum value problems that cannot be solved using the basic inequality.