Finding Critical Values | 求临界值

📚 Finding Critical Values | 求临界值

Critical values are among the most important tools in A-Level calculus. They are the x-values at which the gradient of a curve becomes zero or the derivative fails to exist. In many Edexcel questions, you will be asked to find critical values and then classify them as local maxima, local minima, or stationary inflection points.

临界值是 A-Level 微积分中最重要的工具之一。它们是曲线梯度为零或导数不存在的 x 值。在许多 Edexcel 题目中，你都需要求出临界值，然后将它们分类为局部最大值、局部最小值或驻点拐点。

1. What are critical values? | 什么是临界值

For a function f(x), a critical value is an x-value c in the domain of f such that either f'(c) = 0 or f'(c) is undefined.

对于函数 f(x)，临界值是指在其定义域内使得 f'(c) = 0 或 f'(c) 不存在的 x 值 c。

At such x-values, the tangent to the curve is horizontal, or the curve has a corner, cusp, or vertical tangent.

在这样的 x 值处，曲线的切线是水平的，或者曲线出现角点、尖点或垂直切线。

In Edexcel A-Level Mathematics, the term stationary point is often used when f'(c) = 0. Critical values include stationary points as well as points where differentiation fails.

在 Edexcel A-Level 数学中，当 f'(c) = 0 时常用驻点一词。临界值包括驻点以及导数不存在的点。

2. Why critical values matter | 临界值的重要性

Critical values are essential because local maxima and minima of a

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