📚 Increasing and Decreasing Functions and the Derivative Test | 函数的增减性与导数判定方法
Understanding whether a function is increasing or decreasing is one of the most fundamental applications of differentiation. In A-Level Mathematics, the derivative serves as a powerful tool to determine the monotonic behaviour of a function over an interval, locate turning points, and sketch accurate graphs. This article provides a comprehensive, exam-focused guide to the derivative test for monotonicity.
理解函数在某区间上是递增还是递减，是微积分最基础的应用之一。在A-Level数学中，导数是一个强有力的工具，用于判断函数在区间内的单调性、定位极值点以及精确绘制函数图像。本文将围绕导数判定增减性这一考点，提供全面且贴合考试要求的解析。
1. Definition of Increasing and Decreasing Functions | 函数增减性的定义
Let f(x) be defined on an interval I. We say that f is increasing on I if for any two points x₁ < x₂ in I, we have f(x₁) ≤ f(x₂). If the inequality is strict, the function is called strictly increasing.
设函数 f(x) 在区间 I 上有定义。如果对于区间 I 内任意两点 x₁ < x₂，都有 f(x₁) ≤ f(x₂)，则称 f 在 I 上为递增函数；若不等号严格成立，则称为严格递增。
Similarly, f is decreasing on I if for x₁ < x₂, we have f(x₁) ≥ f(x₂). The term monotonic is used to describe a function that is either entirely increasing or entirely decreasing on an interval.
类似地，如果对于 x₁ < x₂，有 f(x₁) ≥ f(x₂)，则称 f 在 I 上为递减函数。单调一词用于描述函数在整个区间内要么一直递增、要么一直递减的性质。
For x₁ < x₂: Increasing ⇔ f(x₁) < f(x₂) | Decreasing ⇔ f(x₁) > f(x₂)
It is crucial to note that monotonicity is always discussed on a specific interval, not at a single point. A function may increase on one interval and decrease on another, so we must clearly identify the domain intervals under consideration.
需要特别注意的是，单调性总是在某个特定区间上讨论的，而非在某一个点上。同一个函数可能在一个区间递增、在另一个区间递减，因此我们必须明确所考察的区间范围。
2. The Derivative as a Rate of Change | 导数作为变化率
The derivative f‘(x) measures the instantaneous rate of change of f(x) with respect to x. Geometrically, it represents the slope of the tangent line to the curve at a given point. If the slope is positive, the curve rises as x increases; if negative, the curve falls.
导数 f‘(x) 度量的是 f(x) 关于 x 的瞬时变化率。从几何角度看，它代表曲线在给定点处切线的斜率。如果斜率为正，曲线随 x 增大而上升；如果斜率为负，曲线则下降。
Consider a car moving along a straight road. If its position function has a positive derivative, the car is moving forward; a negative derivative means it is moving backward. This intuitive interpretation directly links the sign of the derivative to the direction of motion.
想象一辆汽车沿直线道路行驶。如果其位置函数的导数为正，汽车正在前进；导数为负则意味着汽车在倒退。这种直观的理解将导数的符号与运动方向直接联系起来。
Mathematically, we formalise this connection: if f‘(x) > 0 at every point in an interval, then f is strictly increasing on that interval. If f‘(x) < 0 at every point, then f is strictly decreasing.
在数学上，我们将这种联系形式化：如果在某个区间内每一点都有 f‘(x) > 0，那么 f 在该区间上严格递增；如果每一点都有 f‘(x) < 0，则 f 严格递减。
3. The Derivative Sign Test | 导数符号判定法
The most direct method to determine where a function is increasing or decreasing is to analyse the sign of its derivative. The procedure is straightforward:
判断函数增减性最直接的方法就是分析其导数的符号。具体步骤如下：
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Compute f‘(x) by differentiating f(x).
通过求导计算 f‘(x)。
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Find the critical points by solving f‘(x) = 0 and identify points where f‘(x) is undefined.
解方程 f‘(x) = 0 求出临界点，并找出 f‘(x) 不存在的点。
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Divide the domain into intervals using these critical points.
用这些临界点将定义域划分为若干子区间。
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Test the sign of f‘(x) at one convenient point in each interval.
在每个子区间内选取一个方便的点，测试 f‘(x) 的符号。
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If f‘(x) > 0, the function is increasing; if f‘(x) < 0, it is decreasing.
若 f‘(x) > 0，则函数递增；若 f‘(x) < 0，则函数递减。
This method is often called the first derivative test for monotonicity. It is essential to check the sign on the entire interval — picking a single test point is valid because f‘(x) cannot change sign without passing through zero (or a discontinuity).
这种方法通常被称为判断单调性的一阶导数检验法。需要注意的是，必须检验整个区间上的符号——选取一个测试点是有效的，因为 f‘(x) 要在不经过零点（或不连续点）的情况下改变符号是不可能的。
f‘(x) > 0 ⇒ f increasing | f‘(x) < 0 ⇒ f decreasing
4. Worked Example: Polynomial Function | 例题：多项式函数
Let us apply the derivative sign test to the function f(x) = x³ − 3x² + 2. This classic cubic will help us illustrate every step clearly.
让我们将导数符号判定法应用于函数 f(x) = x³ − 3x² + 2。这个经典的三次函数将帮助我们清晰地展示每一步。
Step 1 – Differentiate:
第一步——求导：
f‘(x) = 3x² − 6x = 3x(x − 2)
Step 2 – Find critical points: Solve f‘(x) = 0, giving 3x(x − 2) = 0, so x = 0 and x = 2.
第二步——求临界点： 解 f‘(x) = 0，得 3x(x − 2) = 0，所以 x = 0 和 x = 2。
Step 3 – Divide the domain: The critical points split the real line into three intervals: (−∞, 0), (0, 2), and (2, ∞).
第三步——划分区间： 临界点将实数轴划分为三个区间：(−∞, 0)、(0, 2) 和 (2, ∞)。
Step 4 – Test the sign: Choose test points x = −1, x = 1, and x = 3.
第四步——测试符号： 选取测试点 x = −1、x = 1 和 x = 3。
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At x = −1: f‘(−1) = 3(−1)(−3) = 9 > 0 → increasing.
在 x = −1 处：f‘(−1) = 3(−1)(−3) = 9 > 0，故递增。
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At x = 1: f‘(1) = 3(1)(−1) = −3 < 0 → decreasing.
在 x = 1 处：f‘(1) = 3(1)(−1) = −3 < 0，故递减。
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At x = 3: f‘(3) = 3(3)(1) = 9 > 0 → increasing.
在 x = 3 处：f‘(3) = 3(3)(1) = 9 > 0，故递增。
Step 5 – Conclude: The function is increasing on (−∞, 0) ∪ (2, ∞) and decreasing on (0, 2). The points x = 0 and x = 2 are stationary points where the tangent is horizontal.
第五步——得出结论： 函数在 (−∞, 0) ∪ (2, ∞) 上递增，在 (0, 2) 上递减。点 x = 0 和 x = 2 是驻点，切线水平。
5. Stationary Points and Turning Points | 驻点与极值点
A stationary point is a point on the curve where f‘(x) = 0. A turning point is a stationary point where the function changes from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum).
驻点是曲线上使得 f‘(x) = 0 的点。极值点（又称转向点）是函数从递增变为递减（局部最大值）或从递减变为递增（局部最小值）的驻点。
It is important to distinguish between these two concepts: every turning point is a stationary point, but not every stationary point is a turning point. For example, the function f(x) = x³ has a stationary point at x = 0, but the function does not turn there — it continues increasing on both sides. This is called a point of inflection with horizontal tangent.
区分这两个概念非常重要：每个极值点都是驻点，但并非每个驻点都是极值点。例如，函数 f(x) = x³ 在 x = 0 处有驻点，但函数在那里并没有转向——它在两侧都持续递增。这种情况称为水平切线的拐点。
To determine whether a stationary point is a local maximum, local minimum, or neither, we examine the sign of the derivative immediately to the left and right of the point. This is the essence of the first derivative test for extrema.
为了判断驻点是局部最大值、局部最小值还是两者都不是，我们需要检查该点左右两侧导数的符号。这就是用于极值判定的一阶导数检验法的核心思想。
| Sign of f‘ to the left | Sign of f‘ to the right | Nature of stationary point |
| + (increasing) | − (decreasing) | Local maximum (局部最大值) |
| − (decreasing) | + (increasing) | Local minimum (局部最小值) |
| + (increasing) | + (increasing) | Point of inflection (拐点) |
| − (decreasing) | − (decreasing) | Point of inflection (拐点) |
6. The Second Derivative Test | 二阶导数检验法
The second derivative f”(x) measures the rate of change of f‘(x), i.e., the curvature of the original function. At a stationary point where f‘(x) = 0, the second derivative provides an alternative way to classify the point:
二阶导数 f”(x) 度量的是 f‘(x) 的变化率，即原函数的弯曲程度。在驻点处（f‘(x) = 0），二阶导数提供了一种备选的极值判定方法：
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If f”(x) > 0, the stationary point is a local minimum (curve is concave up / smiling).
若 f”(x) > 0，驻点为局部最小值（曲线凹向上，呈”微笑”状）。
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If f”(x) < 0, the stationary point is a local maximum (curve is concave down / frowning).
若 f”(x) < 0，驻点为局部最大值（曲线凹向下，呈”皱眉”状）。
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If f”(x) = 0, the test is inconclusive — use the first derivative test instead.
若 f”(x) = 0，检验失效——需要使用一阶导数检验法。
For our earlier example f(x) = x³ − 3x² + 2, we have f”(x) = 6x − 6. At x = 0, f”(0) = −6 < 0, confirming a local maximum; at x = 2, f”(2) = 6 > 0, confirming a local minimum.
对于前述例子 f(x) = x³ − 3x² + 2，我们有 f”(x) = 6x − 6。在 x = 0 处，f”(0) = −6 < 0，确认是局部最大值；在 x = 2 处，f”(2) = 6 > 0，确认是局部最小值。
7. Points of Inflection | 拐点
A point of inflection is a point where the curve changes its concavity — from concave up to concave down, or vice versa. At such a point, the second derivative changes sign. The condition f”(x) = 0 is necessary but not sufficient: we must verify that f” changes sign around the point.
拐点是曲线凹凸性发生改变的点——从凹向上变为凹向下，或相反。在拐点处，二阶导数改变符号。条件 f”(x) = 0 是必要的但非充分的：我们必须验证 f” 在该点两侧确实改变符号。
Consider f(x) = x⁴. We have f”(x) = 12x², and f”(0) = 0, but f”(x) ≥ 0 on both sides of 0. Therefore, x = 0 is not a point of inflection — the curve is concave up everywhere.
考虑函数 f(x) = x⁴。我们有 f”(x) = 12x²，且 f”(0) = 0，但 0 两侧都有 f”(x) ≥ 0。因此，x = 0 不是拐点——曲线处处凹向上。
In contrast, for f(x) = x³, f”(x) = 6x, which is negative for x < 0 and positive for x > 0. Thus, x = 0 is a genuine point of inflection.
相比之下，对于 f(x) = x³，f”(x) = 6x，当 x < 0 时为负，当 x > 0 时为正。因此，x = 0 是真正的拐点。
Inflection condition: f”(x) = 0 and f” changes sign at x
Points of inflection are important in curve sketching because they indicate where the curve stops bending one way and starts bending the other way, creating an ‘S’-shape transition.
拐点在曲线绘图中非常重要，因为它们表示曲线在何处停止向一个方向弯曲并开始向另一个方向弯曲，形成 S 形过渡。
8. Monotonicity and the First Derivative Test in Detail | 单调性与一阶导数检验法详解
The first derivative test for classifying stationary points involves examining the sign of f‘(x) on both sides of a stationary point. Here is the detailed procedure applied to f(x) = x³ − 3x² + 2:
一阶导数检验法通过考察驻点左右两侧 f‘(x) 的符号来对驻点进行分类。以下是应用于 f(x) = x³ − 3x² + 2 的详细步骤：
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At x = 0: For x slightly less than 0 (e.g., x = −0.1), f‘ = 3(−0.1)(−2.1) = 0.63 > 0. For x slightly greater than 0 (e.g., x = 0.1), f‘ = 3(0.1)(−1.9) = −0.57 < 0. Since the sign changes from + to −, we have a local maximum.
在 x = 0 处：取略小于 0 的点（如 x = −0.1），f‘ = 3(−0.1)(−2.1) = 0.63 > 0。取略大于 0 的点（如 x = 0.1），f‘ = 3(0.1)(−1.9) = −0.57 < 0。由于符号从 + 变为 −，这是局部最大值。
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At x = 2: For x slightly less than 2 (e.g., x = 1.9), f‘ = 3(1.9)(−0.1) = −0.57 < 0. For x slightly greater than 2 (e.g., x = 2.1), f‘ = 3(2.1)(0.1) = 0.63 > 0. The sign changes from − to +, indicating a local minimum.
在 x = 2 处：取略小于 2 的点（如 x = 1.9），f‘ = 3(1.9)(−0.1) = −0.57 < 0。取略大于 2 的点（如 x = 2.1），f‘ = 3(2.1)(0.1) = 0.63 > 0。符号从 − 变为 +，表明这是局部最小值。
This sign-change analysis is the most reliable method and works even when f”(x) = 0 at the stationary point, which is precisely where the second derivative test fails.
这种符号变化分析是最可靠的方法，即使在驻点处 f”(x) = 0 时也有效——而正是在这种情况下，二阶导数检验法会失效。
9. Intervals of Increase and Decrease for Exponential and Logarithmic Functions | 指数函数与对数函数的增减区间
Exponential and logarithmic functions play a prominent role in A-Level mathematics. Let us examine their monotonicity systematically.
指数函数和对数函数在A-Level数学中占据重要地位。让我们系统地考察它们的单调性。
For f(x) = e⁻ˣ, the derivative is f‘(x) = −e⁻ˣ. Since e⁻ˣ > 0 for all x, we have f‘(x) < 0 for all x. Therefore, e⁻ˣ is strictly decreasing on the entire real line.
对于 f(x) = e⁻ˣ，其导数为 f‘(x) = −e⁻ˣ。由于对所有 x 都有 e⁻ˣ > 0，所以 f‘(x) < 0 对所有 x 成立。因此，e⁻ˣ 在整个实数轴上严格递减。
For g(x) = ln x (defined for x > 0), the derivative is g‘(x) = 1/x. Since 1/x > 0 for all x > 0, the natural logarithm is strictly increasing on its entire domain (0, ∞).
对于 g(x) = ln x（定义域为 x > 0），其导数为 g‘(x) = 1/x。由于对所有 x > 0 都有 1/x > 0，自然对数在其整个定义域 (0, ∞) 上严格递增。
A useful general result: for f(x) = aˣ with a > 1, the function is always increasing; with 0 < a < 1, it is always decreasing. The derivative f‘(x) = aˣ ln a confirms this, since ln a > 0 when a > 1 and ln a < 0 when 0 < a < 1.
一个有用的一般结论：对于 f(x) = aˣ，当 a > 1 时函数总是递增；当 0 < a < 1 时函数总是递减。导数 f‘(x) = aˣ ln a 验证了这一点，因为当 a > 1 时 ln a > 0，当 0 < a < 1 时 ln a < 0。
10. Common Traps and Misconceptions | 常见陷阱与误区
Several common mistakes appear repeatedly in examinations when students deal with monotonicity and derivatives. Being aware of them is half the battle won.
在处理单调性与导数的题目时，学生在考试中会反复犯一些常见错误。意识到这些误区就等于打赢了一半的战斗。
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Misconception 1: Believing that f‘(x) = 0 implies a local maximum or minimum. In fact, it only gives a stationary point — it could also be a point of inflection with a horizontal tangent, as in f(x) = x³.
误区一：以为 f‘(x) = 0 就意味着局部最大值或最小值。实际上，这只是给出了一个驻点——它也可能是具有水平切线的拐点，如 f(x) = x³。
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Misconception 2: Confusing functions that are increasing on an interval with functions whose derivative is never zero. A function can be strictly increasing even if f‘(x) = 0 at isolated points — for example, f(x) = x³ is strictly increasing on ℝ, yet f‘(0) = 0.
误区二：把函数在区间上递增与导数永不为零混为一谈。即使 f‘(x) 在某些孤立点处为零，函数仍然可以严格递增——例如，f(x) = x³ 在 ℝ 上严格递增，但 Published by TutorHao | Mathematics Revision Series | aleveler.com
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