📚 Determining Monotonicity and Intervals of Increase/Decrease | 函数单调性与增减区间判定
In A-Level Mathematics, understanding whether a function is increasing or decreasing on a given interval is essential for sketching curves, locating extrema and solving optimisation problems. The most powerful tool is the first derivative.
在A-Level数学中，理解函数在给定区间上是递增还是递减，对于绘制曲线、寻找极值和解决优化问题至关重要。最有力的工具是一阶导数。
1. Definition of Increasing and Decreasing Functions | 增函数与减函数的定义
A function f(x) is said to be increasing on an interval (a,b) if for any x₁ < x₂ in (a,b), we have f(x₁) ≤ f(x₂). It is strictly increasing if f(x₁) < f(x₂). Similarly, f is decreasing on (a,b) if f(x₁) ≥ f(x₂) for all x₁ < x₂, and strictly decreasing if f(x₁) > f(x₂).
函数 f(x) 在区间 (a,b) 上被称为递增的，如果对于 (a,b) 内任意 x₁ < x₂，都有 f(x₁) ≤ f(x₂)。若 f(x₁) < f(x₂)，则称为严格递增。类似地，f 在 (a,b) 上递减是指对所有 x₁ < x₂ 有 f(x₁) ≥ f(x₂)；若 f(x₁) > f(x₂)，则称为严格递减。
2. The Role of the First Derivative | 一阶导数的作用
If f is differentiable on (a,b), the sign of the derivative f'(x) determines the monotonicity. If f'(x) > 0 for all x in (a,b), then f is strictly increasing on that interval. If f'(x) < 0 for all x in (a,b), then f is strictly decreasing. If f'(x) = 0 for all x, the function is constant.
若 f 在 (a,b) 上可导，则导数 f'(x) 的符号决定其单调性。若在 (a,b) 内恒有 f'(x) > 0，则 f 在该区间上严格递增；若恒有 f'(x) < 0，则 f 严格递减；若 f'(x) = 0，则函数为常函数。
In particular, when the derivative is positive, small increases in x produce positive changes in f(x). When the derivative is negative, the function values fall as x rises. This relationship is the foundation of all interval analysis.
特别地，当导数为正时，x 的微小增加会导致 f(x) 正向变化；当导数为负时，函数值随 x 增大而下降。这一关系是所有区间分析的基础。
3. Critical Points and Stationary Points | 临界点与驻点
A critical point of f occurs where f'(x) = 0 or where the derivative does not exist. A stationary point is a critical point with f'(x) = 0. These points are the only candidates where the sign of the derivative can change, so they divide the domain into test intervals.
函数 f 的临界点出现在 f'(x) = 0 或导数不存在处。驻点是 f'(x) = 0 的临界点。这些点是导数符号可能发生变化的唯一候选点，因此它们将定义域分割成若干测试区间。
Set the derivative equal to zero:
令导数为零：
f'(x) = 0
Solutions are called stationary points. Points where f'(x) is undefined may be corners, cusps or vertical tangents; they must also be considered as potential boundaries of monotonic intervals.
该方程的解称为驻点。f'(x) 不存在的点可能是尖点、角点或竖直切线；它们也必须被视为单调区间的潜在分界点。
4. Procedure for Finding Intervals | 求增减区间的步骤
To find intervals of increase and decrease, follow these steps:
为求增减区间，可按以下步骤操作：
1. Determine the domain of f(x).
2. Compute f'(x).
3. Solve f'(x) = 0, and note points where f’ is undefined.
4. Place these points on a number line and choose test values in each interval.
5. Evaluate the sign of f’ at each test value.
6. Conclude where f’ > 0 (increasing) and where f’ < 0 (decreasing).
1. 确定 f(x) 的定义域。
2. 计算 f'(x)。
3. 解方程 f'(x) = 0，并注意 f’ 不存在的点。
4. 将这些点在数轴上标出，并在每个区间内选取测试值。
5. 计算每个测试点处 f’ 的符号。
6. 由此得出 f’ > 0（递增）和 f’ < 0（递减）的区间。
5. Sign Diagram of the Derivative | 导数的符号图
A sign diagram is a number line showing the domain divided by critical points, with + or − symbols indicating the sign of f’ in each interval. It is a compact way to record the analysis and is often expected in exam solutions.
符号图是一条被临界点分割的数轴，用 + 或 − 表示每个区间内 f’ 的符号。这是一种紧凑的记录分析方式，也是考试中常用并要求写出的答案形式。
For a quadratic example, the sign diagram may look like this:
对于二次函数的例子，符号图可能如下所示：
| x | (−∞, 2) | x = 2 | (2, ∞) |
| f'(x) | − | 0 | + |
Always include the critical values on the number line, but write the final intervals as open intervals unless the function is defined and continuous at the endpoints.
在数轴上务必标出临界值，但最终区间应写成开区间，除非函数在端点处有定义且连续。
6. Worked Example 1: Quadratic Function | 例1：二次函数
Find the intervals of increase and decrease for f(x) = x² − 4x + 3. The derivative is f'(x) = 2x − 4.
求 f(x) = x² − 4x + 3 的增减区间。导数为 f'(x) = 2x − 4。
Setting f'(x) = 0 gives x = 2. Test x = 0: f'(0) = −4 < 0, so f is decreasing on (−∞, 2). Test x = 3: f'(3) = 2 > 0, so f is increasing on (2, ∞).
令 f'(x) = 0 得 x = 2。取 x = 0 测试，f'(0) = −4 < 0，所以 f 在 (−∞, 2) 上递减；取 x = 3 测试，f'(3) = 2 > 0，所以 f 在 (2, ∞) 上递增。
7. Worked Example 2: Cubic Function | 例2：三次函数
Consider f(x) = x³ − 3x + 1. Then f'(x) = 3x² − 3 = 3(x − 1)(x + 1). The critical points are x = −1 and x = 1.
考虑 f(x) = x³ − 3x + 1，则 f'(x) = 3x² − 3 = 3(x − 1)(x + 1)。临界点为 x = −1 和 x = 1。
Choose test values: x = −2 gives f'(−2) = 9 > 0; x = 0 gives f'(0) = −3 < 0; x = 2 gives f'(2) = 9 > 0. Therefore f is increasing on (−∞, −1) ∪ (1, ∞) and decreasing on (−1, 1).
选择测试值：x = −2 时 f'(−2) = 9 > 0；x = 0 时 f'(0) = −3 < 0；x = 2 时 f'(2) = 9 > 0。因此 f 在 (−∞, −1) ∪ (1, ∞) 上递增，在 (−1, 1) 上递减。
8. Worked Example 3: Rational Function | 例3：有理函数
Let f(x) = 1/x. The derivative is f'(x) = −1/x². Since x² > 0 for x ≠ 0, f'(x) < 0 for all x in its domain.
设 f(x) = 1/x，导数为 f'(x) = −1/x²。由于 x ≠ 0 时 x² > 0，所以定义域内 f'(x) < 0。
Therefore f is decreasing on (−∞, 0) and also on (0, ∞), but not on the union because the function has a discontinuity at x = 0. The two intervals must be treated separately.
因此 f 在 (−∞, 0) 和 (0, ∞) 上分别递减，但不能说在整个并集上递减，因为函数在 x = 0 处不连续。这两个区间必须分别处理。
9. Discontinuities and Non-differentiable Points | 不连续点与不可导点
When the derivative is undefined at a point such as a corner or vertical tangent, the sign of f’ may change there without a stationary point. For rational functions, vertical asymptotes also split the domain; always test each connected component separately.
当函数在某点不可导（例如尖点或竖直切线）时，f’ 的符号可能在该点发生变化，即使没有驻点。对于有理函数，垂直渐近线也会分割定义域；务必对每个连通部分分别测试。
A common example is f(x) = |x|. The derivative is −1 for x < 0 and +1 for x > 0, but it is undefined at x = 0. Thus f decreases on (−∞, 0) and increases on (0, ∞).
常见例子是 f(x) = |x|。当 x < 0 时导数为 −1，x > 0 时导数为 +1，但在 x = 0 处不可导。因此 f 在 (−∞, 0) 上递减，在 (0, ∞) 上递增。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common errors include: forgetting to state the domain; testing only one point per interval but choosing a point equal to a critical value; writing intervals with square brackets at endpoints where the derivative is zero or undefined; and assuming that f’ > 0 at every point implies the function is monotonic even if f is not continuous.
常见错误包括：忘记给出定义域；每个区间只测一个点但该点恰为临界值；在导数为零或不存在处用闭区间；以及假定 f’ > 0 处处成立就一定单调，而忽略函数连续性。
In Edexcel exams, always show the derivation of f'(x), solve f'(x) = 0 explicitly, draw a sign diagram, and state the final intervals clearly. Use open intervals for the monotonicity unless the function is also increasing on the closed interval when continuity is guaranteed.
在 Edexcel 考试中，务必写出求导过程，明确解 f'(x) = 0，画出符号图，并清晰地陈述最终区间。除非连续性保证函数在闭区间上仍递增，否则单调区间应使用开区间。
11. Summary | 总结
In summary, the sign of the first derivative fully determines where a differentiable function is increasing or decreasing. Always work with the function’s domain, find all critical points, test each interval, and present your conclusion using interval notation.
总之，一阶导数的符号完全确定了可导函数在何处递增、何处递减。处理时务必考虑函数定义域，找出所有临界点，测试每个区间，并用区间记号给出结论。
With practice, sign analysis becomes quick and reliable. It is one of the most transferable skills in calculus and appears in many examination contexts.
通过练习，符号分析会变得快速而可靠。这是微积分中最具迁移性的技能之一，在考试中频繁出现。
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