Determining Increase and Decrease of Functions in A-Level Maths | A-Level数学：函数增减性的判断方法

📚 Determining Increase and Decrease of Functions in A-Level Maths | A-Level数学：函数增减性的判断方法

Understanding whether a function is increasing or decreasing over an interval is a fundamental skill in A-Level Mathematics. It forms the backbone of curve sketching, optimisation problems, and interpreting real-world rates of change. This guide introduces the core methods used to determine monotonicity, with a focus on the derivative approach that examiners expect.

判断函数在某个区间内是递增还是递减，是A-Level数学中的基础技能。它贯穿于曲线作图、最优化问题以及实际变化率的理解之中。本文将系统介绍判断函数增减性的核心方法，重点讲解考试中必不可少的导数法。

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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 if, for any two points x₁ and x₂ within that interval with x₁ < x₂, we have f(x₁) ≤ f(x₂). It is strictly increasing if f(x₁) < f(x₂). Similarly, f(x) is decreasing if f(x₁) ≥ f(x₂), and strictly decreasing if f(x₁) > f(x₂).

函数 f(x) 在某个区间上被称为递增的，如果对于区间内任意两点 x₁ 和 x₂，当 x₁ < x₂ 时，总有 f(x₁) ≤ f(x₂)。若不等式严格成立，即 f(x₁) < f(x₂)，则称严格递增。类似地，若 f(x₁) ≥ f(x₂) 则称递减，若 f(x₁) > f(x₂) 则称严格递减。

These definitions are the starting point for all reasoning about monotonicity. In practice, however, checking every pair of points is impractical — which is where calculus comes in.

这些定义是所有单调性推理的出发点。不过在实践中，逐一检查每一对点并不现实——这正是微积分发挥作用的地方。

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2. The Derivative Test: f'(x) > 0 or f'(x) < 0 | 导数判别法：f'(x) > 0 或 f'(x) < 0

The most powerful and frequently used method in A-Level is the derivative test. If a function f(x) is differentiable on an open interval (a, b), then:

A-Level中最常用且最有力的方法就是导数判别法。若函数 f(x) 在开区间 (a, b) 内可导，则：

• If f'(x) > 0 for all x in (a, b), then f(x) is strictly increasing on (a, b).
• 若在 (a, b) 内对所有 x 都有 f'(x) > 0，则 f(x) 在 (a, b) 上严格递增。
• If f'(x) < 0 for all x in (a, b), then f(x) is strictly decreasing on (a, b).
• 若在 (a, b) 内对所有 x 都有 f'(x) < 0，则 f(x) 在 (a, b) 上严格递减。
• If f'(x) = 0 for all x in (a, b), then f(x) is constant on (a, b).
• 若在 (a, b) 内对所有 x 都有 f'(x) = 0，则 f(x) 在 (a, b) 上是常数函数。

f'(x) > 0 → increasing | f'(x) < 0 → decreasing | f'(x) = 0 → constant

The derivative measures the instantaneous rate of change. A positive rate means the function value rises as x increases; a negative rate means it falls. This geometric interpretation makes the test intuitive and easy to apply.

导数衡量的是瞬时变化率。变化率为正意味着函数值随 x 增大而上升；变化率为负则意味着函数值随 x 增大而下降。这种几何意义使得导数判别法既直观又易于运用。

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3. Worked Example: Polynomial Function | 实例分析：多项式函数

Consider the function f(x) = x³ − 3x + 2. Let us determine the intervals where it is increasing or decreasing.

考虑函数 f(x) = x³ − 3x + 2。我们来判断它在哪些区间上递增、哪些区间上递减。

Step 1: Differentiate.

第一步：求导。

f'(x) = 3x² − 3 = 3(x − 1)(x + 1)

Step 2: Solve f'(x) = 0 to find critical points.

第二步：解 f'(x) = 0 找出临界点。

3(x − 1)(x + 1) = 0 → x = 1 or x = −1

Step 3: Test the sign of f'(x) in each interval determined by these critical points.

第三步：在各个由临界点划分的区间内测试 f'(x) 的符号。

Interval	Test value	Sign of f'(x)	Conclusion
x < −1	x = −2	(+) × (−) × (−) = (+)	Increasing ▲
−1 < x < 1	x = 0	(+) × (−) × (+) = (−)	Decreasing ▼
x > 1	x = 2	(+) × (+) × (+) = (+)	Increasing ▲

Notice how the sign test works: on each interval, choose a convenient test value and substitute it into the factorised derivative. This avoids expanding and re-factorising.

注意符号测试的技巧：在每个区间选择一个方便的测试点，代入因式分解后的导数表达式中。这样可以避免展开后再重新分解的步骤。

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4. Interpreting Stationary Points | 驻点的含义

Points where f'(x) = 0 are called stationary points. They are the boundaries between intervals of increase and decrease. At a stationary point, the function momentarily stops rising or falling.

使 f'(x) = 0 的点称为驻点。它们是递增区间与递减区间的分界点。在驻点处，函数的上升或下降趋势会暂时停止。

In the example above, x = −1 is a local maximum and x = 1 is a local minimum, determined by the sign change of f'(x): positive → negative at x = −1, and negative → positive at x = 1.

在上面的例子中，x = −1 是局部极大值点，x = 1 是局部极小值点。这是由 f'(x) 的符号变化判断的：在 x = −1 处由正变负，在 x = 1 处由负变正。

• If f'(x) changes from + to − at a stationary point → local maximum.
• 若 f'(x) 在驻点处由 + 变为 − → 局部极大值。
• If f'(x) changes from − to + at a stationary point → local minimum.
• 若 f'(x) 在驻点处由 − 变为 + → 局部极小值。
• If f'(x) does not change sign → point of inflection (horizontal).
• 若 f'(x) 符号不变 → 拐点（水平拐点）。

This sign-change analysis is a core examiner requirement — always state the sign pattern explicitly in your working.

这种符号变化分析是考官的核要求——在解答过程中务必明确写出符号模式。

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5. Using the Definition Directly | 直接用定义判断

When a function is not easily differentiable, or when an exam question explicitly asks for a definition-based proof, you must fall back on the formal definition. For example, prove that f(x) = 2x + 1 is strictly increasing on ℝ.

当函数不便求导，或考题明确要求基于定义证明时，就必须回归形式化定义。例如，证明 f(x) = 2x + 1 在 ℝ 上严格递增。

Take any x₁ < x₂. Then:

任取 x₁ < x₂，则：

x₂ − x₁ > 0 ⇒ f(x₂) − f(x₁) = (2x₂ + 1) − (2x₁ + 1) = 2(x₂ − x₁) > 0

Since f(x₂) − f(x₁) > 0, we have f(x₂) > f(x₁), which by definition means f is strictly increasing.

因为 f(x₂) − f(x₁) > 0，即 f(x₂) > f(x₁)，根据定义，f 是严格递增的。

For rational functions like f(x) = 1/x, you can use a similar approach over intervals like (0, ∞), being careful about sign of x₁ and x₂.

对于有理函数如 f(x) = 1/x，可以在 (0, ∞) 等区间上采用类似方法，但需特别注意 x₁ 与 x₂ 的符号。

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6. Graphical Interpretation | 图像法判断

Sometimes the fastest way to determine monotonicity is to sketch the curve or use a graphical calculator. Reading the graph from left to right:

有时最快速的方法就是画出曲线草图或使用图形计算器。从左向右读图：

• If the curve goes upward as x increases → increasing.
• 若曲线随 x 增大而上升 → 递增。
• If the curve goes downward as x increases → decreasing.
• 若曲线随 x 增大而下降 → 递减。
• Horizontal segments indicate constant regions.
• 水平线段表示常数区域。

The derivative test is essentially the analytic version of this visual inspection. However, graphs can mislead when features are small or when the domain is restricted — always combine graphical intuition with algebraic verification.

导数判别法本质上就是视觉判断的解析版本。不过，当图像特征较小或定义域受限时，图形可能会造成误导——务必结合图形直觉与代数验证。

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7. Composite Functions and Chain Rule | 复合函数与链式法则

For composite functions h(x) = g(f(x)), the monotonicity depends on both the inner function f and the outer function g. Using the chain rule:

对于复合函数 h(x) = g(f(x))，其增减性取决于内层函数 f 和外层函数 g。根据链式法则：

h'(x) = g'(f(x)) · f'(x)

The sign of h'(x) is the product of two signs. Therefore:

h'(x) 的符号是两个符号的乘积。因此：

• If both g'(f(x)) and f'(x) have the same sign → h is increasing.
• 若 g'(f(x)) 与 f'(x) 同号 → h 递增。
• If they have opposite signs → h is decreasing.
• 若二者异号 → h 递减。

A common A-Level example is h(x) = e^(x²). Here f(x) = x² and g(u) = eᵘ. On x > 0, f'(x) = 2x > 0 and g'(u) = eᵘ > 0, so h'(x) > 0 — h is increasing for x > 0.

一个常见的A-Level例子是 h(x) = e^(x²)。这里 f(x) = x²，g(u) = eᵘ。在 x > 0 时，f'(x) = 2x > 0，且 g'(u) = eᵘ > 0，所以 h'(x) > 0 —— 即 h 在 x > 0 时递增。

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8. Inverse Functions and Monotonicity | 反函数与增减性

If a function f is strictly increasing on its domain, then its inverse f⁻¹ is also strictly increasing. If f is strictly decreasing, then f⁻¹ is also strictly decreasing.

如果函数 f 在其定义域上严格递增，那么它的反函数 f⁻¹ 也是严格递增的。如果 f 严格递减，则 f⁻¹ 也严格递减。

The reason is geometric: the graph of f⁻¹ is the reflection of f across the line y = x, and a reflection preserves the direction of slope.

其几何原因在于：f⁻¹ 的图像是 f 的图像关于直线 y = x 的反射，而反射保持斜率的升降方向不变。

• f(x) = eˣ is strictly increasing on ℝ → f⁻¹(x) = ln x is strictly increasing on (0, ∞).
• f(x) = eˣ 在 ℝ 上严格递增 → f⁻¹(x) = ln x 在 (0, ∞) 上严格递增。
• f(x) = x³ is strictly increasing on ℝ → f⁻¹(x) = x^(1/3) is strictly increasing on ℝ.
• f(x) = x³ 在 ℝ 上严格递增 → f⁻¹(x) = x^(1/3) 在 ℝ 上严格递增。

This property is occasionally tested directly; remember it also guarantees the existence of an inverse when monotonicity is strict over the whole domain.

这个性质偶尔会被直接考查；同时要记住，当函数在整个定义域上严格单调时，反函数的存在性也能得到保证。

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9. Common Mistakes and Pitfalls | 常见错误与易错点

Even strong students make predictable errors in monotonicity problems. Here are the top ones to avoid:

即使是优秀的学生也会在增减性问题中犯一些可预见的错误。以下是最常见的几点，务必避免：

• Applying the derivative test at a single point instead of over an interval. Monotonicity is an interval property, not a point property.
• 只在单个点使用导数判别法，而不是在整个区间上。单调性是区间性质，不是点性质。
• Forgetting to exclude points where f'(x) = 0 when claiming strict monotonicity.
• 在说明严格单调性时，忘记排除使 f'(x) = 0 的点。
• Dividing by a variable expression without checking its sign — doing so can flip inequalities.
• 在未检查符号的情况下除以含变量的表达式——这可能使不等式反向。
• Confusing “decreasing” with “negative function values”. A function can be decreasing while remaining positive.
• 混淆“递减”与“函数值为负”。一个函数完全可以在保持正数的同时递减。
• Neglecting the domain when the function is defined piecewise or has asymptotes.
• 当函数分段定义或存在渐近线时，忽略定义域问题。

Always write the interval explicitly and check that the function is continuous and differentiable on that interval before applying calculus methods.

在应用微积分方法之前，务必明确写出区间，并检查函数在该区间上的连续性与可导性。

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10. Summary and Exam Strategy | 总结与考试策略

To determine whether a function is increasing or decreasing:

要判断一个函数的增减性：

Step	Action
1	Differentiate f(x) to obtain f'(x).
2	Solve f'(x) = 0 to find stationary points.
3	Divide the domain into intervals using these points.
4	Test the sign of f'(x) in each interval.
5	State the intervals of increase/decrease clearly.

In an exam, always show the factorised form of f'(x), state the critical values, and present the sign table — examiners reward clear structured working.

在考试中，务必展示 f'(x) 的因式分解形式，写出临界值，并给出符号表格——考官对条理清晰的解题过程会给予明确加分。

Mastering these methods will not only earn you full marks on monotonicity questions but also build a strong foundation for curve sketching, integration applications, and optimisation — all of which are heavily represented in A-Level papers.

熟练掌握这些方法，不仅能让你在增减性问题中拿满分数，更为曲线作图、积分应用和最优化问题奠定坚实基础——而这些考点在A-Level试卷中占据着重要比重。

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