📚 Finding and Classifying Stationary Points | 函数驻点的求解与判定
In Edexcel IGCSE Mathematics, one of the most frequently tested skills in the calculus topic is finding the stationary points of a function and deciding whether each one is a local maximum or a local minimum. This technique appears in algebra questions, curve sketching and practical optimisation problems. Whether you are aiming for A* or simply strengthening your revision notes, mastering this method accurately is essential. In this guide, we will work through the complete process: how to locate stationary points, how to classify them using the second derivative or a sign table, and how to apply these ideas to exam-style questions.
在 Edexcel IGCSE 数学中，微积分部分最常考查的技巧之一，就是求函数的驻点，并判断该点是局部极大值还是局部极小值。这一技能出现在代数题、草图绘制和实际优化问题中。无论你的目标是 A*，还是想要巩固复习笔记，准确掌握这一方法都至关重要。在本指南中，我们将完整地讲解整个过程：如何求出驻点，如何用二阶导数或符号表判断其性质，以及如何将这些思路应用到考试风格的题目中。
1. What Are Stationary Points? | 什么是驻点
A stationary point is a point on a curve where the gradient of the tangent is zero. In calculus language, this means the first derivative of the function is zero at that point.
驻点是指曲线上切线梯度（斜率）为零的点。用微积分的语言来说，就是函数在该点的一阶导数为零。
If we write the function as y = f(x), then at a stationary point:
若函数写作 y = f(x)，则在驻点处满足：
dy/dx = 0 或 f'(x) = 0
On a graph, the tangent line at a stationary point is horizontal. The curve may reach a maximum point, a minimum point, or a stationary point of inflection. In the IGCSE syllabus, you will mainly be asked to distinguish between maximum and minimum points.
在图像上，驻点处的切线是水平的。曲线可能在驻点处达到极大值点、极小值点，或者水平拐点。在 IGCSE 考纲中，主要要求大家区分极大值点和极小值点。
2. Quick Review: Differentiating Polynomials | 快速回顾：多项式的求导
Before finding stationary points, you must be confident in differentiating polynomial terms. The single most important rule is the power rule:
在求驻点之前，你必须要熟练对多项式各项求导。其中最重要的一条规则就是幂函数求导法则：
y = axⁿ → dy/dx = anxⁿ⁻¹
For example, if y = 5x⁴, then dy/dx = 5 × 4x³ = 20x³. Constant terms have derivative zero: if y = 7, then dy/dx = 0. This rule is used repeatedly when solving dy/dx = 0.
例如，若 y = 5x⁴，则 dy/dx = 5 × 4x³ = 20x³。常数项的导数为零：若 y = 7，则 dy/dx = 0。在解 dy/dx = 0 时，这个法则会反复使用。
3. Finding Stationary Points Step by Step | 分步求解驻点
The process of finding stationary points can be summarised in four clear steps:
求驻点的过程可以归纳为清晰的四个步骤：
- Step 1: Differentiate the function to find dy/dx.
- Step
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