📚 Increasing and Decreasing Functions | 增函数与减函数
Understanding where a function rises and falls is a core part of calculus. In IB Mathematics, the concepts of increasing and decreasing functions allow us to sketch accurate graphs, solve optimisation problems, and interpret real-world rates of change. This article covers definitions, first derivative tests, interval analysis, and common mistakes – all with clear English–Chinese paired explanations.
理解函数的上升与下降区间是微积分的核心内容。在IB数学中，掌握增函数与减函数的概念有助于准确绘制图像、解决最优化问题并解释现实中的变化率。本文涵盖定义、一阶导数检验、区间分析及常见错误，全部采用清晰的中英对照讲解。
1. What Are Monotonic Functions? | 什么是单调函数？
A function is called monotonic if its graph consistently moves in one direction – either always increasing or always decreasing over an interval. In calculus, we prefer a more precise definition based on the derivative, but the intuitive idea is simple: as x-values get larger, do the y-values go up or down? Monotonicity is the foundation for analysing any function's behaviour.
如果一个函数的图像始终朝一个方向走——在某一区间内总是上升或者总是下降，就称该函数是单调的。在微积分中，我们更倾向于用导数给出精确定义，但直观的概念很简单：随着x值变大，y值是上升还是下降？单调性是分析任何函数行为的基础。
2. Definition of an Increasing Function | 增函数的定义
wo numbers x₁ and x₂ in I with x₁ < x₂, we have f(x₁) ≤ f(x₂). If the inequality is strict – f(x₁) < f(x₂) – to right, the curve goes uphill. In IB problems, we often use the derivative condition: f'(x) > 0 on an interval implies f is strictly increasing there.
对于区间I上的函数f(x)，如果对于I中任意两点x₁ < x₂，都有f(x₁) ≤ f(x₂)，则称f(x)在该区间上是增函数。如果不等式是严格的，即f(x₁) < f(x₂)，则为严格增函数。从图像上看，从左向右移动时曲线向上攀升。在IB问题中，我们通常使用导数条件：在区间上f'(x) > 0意味着f在该区间严格递增。
3. Definition of a Decreasing Function | 减函数的定义
(x₂). It is strictly decreasing when f(x₁) > f(x₂). Visually, the graph moves downhill as x increases. The commonly used derivative criterion is f' decreasing function. Memorising these sign relationships is essential for calculus exams.
对于区间I上的函数f(x)，如果对于I中任意两点x₁ < x₂，都有f(x₁) ≥ f(x₂)，则称f(x)在该区间上是减函数。当f(x₁) > f(x₂)时为严格减函数。从图像上看，随着x增大，图像向下走。常用的导数判断标准是：严格减函数满足f'(x) < 0。牢记这些符号关系对于微积分考试至关重要。
4. Strictly vs Non-Strict Monotonicity | 严格单调与非严格单调
focuses on strict monotonicity, but it is worth knowing the difference. A function with f'(x) ≥ 0 everywhere on an interval might still be strictly increasing unless f'(x) = 0 on a whole subinterval. For example, f(x) = x³ has f'(0) = 0 yet is strictly increasing on ℝ creasing does not. When testing with the derivative, we usually look for intervals where f'(x) > 0 (strict increase) or f'(x) < 0 (strict decrease) and treat points where f'(x) = 0 as boundaries.
IB通常重点关注严格单调，但了解区别也有益。如果一个函数在某区间上处处满足 f'(x) ≥ 0，它仍可能是严格增函数，除非 f'(x) = 0 在一整个子区间上恒成立。例如 f(x) = x³ 在 x=0 处导数为零，但在全体实数上仍是严格增函数。非严格递增允许出现平台段，而严格递增则不允许。用导数检验时，我们通常寻求 f'(x) > 0（严格增）或 f'(x) < 0（严格减）的区间，并把导数等于零的点视为分界点。
5. The First Derivative Test for Monotonicity | 单调性的一阶导数检验
The sign of the first derivative tells us whether a function is rising or falling. The rule is straightforward:
- f'(x) > 0 for all x in (a, b) ⇒ f is strictly increasing on [a, b].
- f'(x) < 0 for all x in (a, b) ⇒ f is strictly decreasing on [a, b].
- If f'(x) = 0 for all x in an interval, f is constant on that interval.
This test works because the derivative measures the instantaneous rate of change. Positive slope means the tangent line tilts upward, so the function must be going up locally, and if that holds throughout an interval, the whole function increases.
一阶导数的正负号告诉我们函数正在上升还是下降。规则非常简单：
- 在 (a, b) 内恒有 f'(x) > 0 ⇒ f 在 [a, b] 上严格递增。
- 在 (a, b) 内恒有 f'(x) < 0 ⇒ f 在 [a, b] 上严格递减。
- 若在某个区间内 f'(x) = 0 恒成立，则 f 在该区间上为常数。
这个检验之所以有效，是因为导数衡量的是瞬时变化率。斜率为正意味着切线向上倾斜，因此函数在该点附近必定上升；如果这一情况在整个区间内都成立，整个函数就是递增的。
6. Finding Intervals of Increase and Decrease | 寻找递增与递减区间
To determine where a function increases or decreases, follow a systematic procedure:
- Compute the derivative f'(x).
- Solve f'(x) = 0 to find critical points. (Also include points where f'(x) does not exist but f is defined.)
- Use these critical points to split the real number line into subintervals.
- Pick a test value from each subinterval and evaluate the sign of f'(x).
- Conclude the behaviour on each interval: positive sign means increasing, negative sign means decreasing.
This sign‑chart method is a favourite in IB exams because it organises your work clearly.
要确定函数的递增和递减区间，可遵循以下系统步骤：
- 计算导数 f'(x)。
- 解方程 f'(x) = 0 找出临界点。（同时也要考虑 f'(x) 不存在但 f 有定义的点。）
- 利用这些临界点将实数线分割成若干子区间。
- 在每个子区间内选取一个测试值，代入 f'(x) 判断符号。
- 得出各区间上的行为结论：符号为正表示递增，为负表示递减。
这种符号表格法在IB考试中很受欢迎，因为它能将解题过程组织得十分清晰。
7. Critical Points and Sign Charts | 临界点与符号表
Critical points occur where f'(x) = 0 or where f' is a number line with critical points marked; above each interval, we write the sign of f'(x). For example, if f'(x) = 3x² - 12, set 3x² - 12 = 0 gives x = ±2. Testing values yields:
| Interval | Test x | Sign of f'(x) | Behaviour |
|---|---|---|---|
| (-∞, -2) | -3 | + | Increasing |
| (-2, 2) | 0 | - | Decreasing |
| (2, ∞) | 3 | + | Increasing |
临界点出现在 f'(x) = 0 或 f'(x) 无定义（但 f 必须在该点有定义）之处。这些是导数符号唯一可能发生变化的地方。符号表是一条标有临界点的数轴；在每个区间上方写出 f'(x) 的符号。例如，若 f'(x) = 3x² - 12，令 3x² - 12 = 0 得 x = ±2。代入测试值后得到：
| 区间 | 测试 x | f'(x) 符号 | 行为 |
|---|---|---|---|
| (-∞, -2) | -3 | + | 递增 |
| (-2, 2) | 0 | - | 递减 |
| (2, ∞) | 3 | + | 递增 |
8. Worked Example: Polynomial Function | 例题：多项式函数
Find the intervals where f(x) = x⁴ - 8x² + 3 is increasing or decreasing.
Solution:
Compute derivative: f'(x) = 4x³ - 16x = 4x(x² - 4) = 4x(x - 2)(x + 2).
Set f'(x) = 0 ⇒ x = -2, 0, 2.
Create a sign chart with four intervals: (-∞, -2), (-2, 0), (0, 2), (2, ∞).
sitive → increasing; x = 1 gives negative → decreasing; x = 3 gives positive → increasing.
Thus, f is increasing on (-2, 0) ∪ (2, ∞) and decreasing on (-∞, -2) ∪ (0, 2).
求函数 f(x) = x⁴ - 8x² + 3 的递增和递减区间。
解：
求导：f'(x) = 4x³ - 16x = 4x(x² - 4) = 4x(x - 2)(x + 2)。
令 f'(x) = 0 得 x = -2, 0, 2。
构建符号表，得到四个区间：(-∞, -2), (-2, 0), (0, 2), (2, ∞)。
测试值：x = -3 时乘积为负 → 递减；x = -1 时为正 → 递增；x = 1 时为负 → 递减；x = 3 时为正 → 递增。
因此，f 在 (-2, 0) ∪ (2, ∞) 上递增，在 (-∞, -2) ∪ (0, 2) 上递减。
9. Worked Example: Rational Function | 例题：有理函数
Determine monotonicity intervals for g(x) = x / (x² + 1). Begin with the quotient rule:
g'(x) = (1·(x²+1) - x·(2x)) / (x²+1)² = (1 - x²) / (x²+1)².
The denominator is always positive, so the sign of g' depends on numerator 1 - x². Set 1 - x² = 0 → x = ±1.
Test intervals: (-∞, -1) pick x = -2 → 1 - 4 = -3 < 0 ⇒ decreasing;
(-1, 1) pick x = 0 → 1 - 0 > 0 ⇒ increasing;
(1, ∞) pick x = 2 → 1 - 4 < 0 ⇒ decreasing.
Hence, g is increasing on (-1, 1) and decreasing on (-∞, -1) ∪ here are no vertical asymptotes affecting the sign because the denominator never vanishes.
判断函数 g(x) = x / (x² + 1) 的单调区间。先用商的求导法则：
g'(x) = (1·(x²+1) - x·(2x)) / (x²+1)² = (1 - x²) / (x²+1)²。
分母恒正，因此 g' 的符号取决于分子 1 - x²。令 1 - x² = 0 得 x = ±1。
测试区间：(-∞, -1) 选取 x = -2 → 1 - 4 = -3 < 0 ⇒ 递减；
(-1, 1) 选取 x = 0 → 1 - 0 > 0 ⇒ 递增；
(1, ∞) 选取 x = 2 → 1 - 4 < 0 ⇒ 递减。
因此，g 在 (-1, 1) 上递增，在 (-∞, -1) ∪ (1, ∞) 上递减。需注意，因为分母永不为零，不存在影响符号的垂直渐近线。
10. Graphical Interpretation | 图像解读
On a function's graph, increasing intervals appear as sections where the curve climbs as you go rightwards. Decreasing intervals look like downhill slides. At the transition points (critical points), the curve may have a peak, a trough, or a horizontal inflection. Connecting the monotonicity with the shape of f helps you sketch accurate graphs without plotting every point. Always mark the intervals of increase and decrease below the x-axis on your sign chart, and transfer them to a rough graph.
在函数图像上，递增区间表现为曲线随向右移动而上升的部分，递减区间则像是下坡路段。在转折点（临界点）处，曲线可能出现峰顶、谷底或水平拐点。将单调性与 f 的形状联系起来，有助于无需逐点描画便能勾勒出准确图像。务必在符号表数轴下方标注增减区间，并将它们转移到草图上去。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与规避方法
- Forgetting the domain: , the monotonicity intervals must break at a, even if the derivative sign does not change.
- Confusing increasing with positive derivative: f'(x) > 0 guarantees strict increase, but a function can be increasing even if f'(x) = 0 at isolated points (e.g., x³ at x=0). The correct condition is f'(x) ≥ 0 and not identically zero on any subinterval.
- Sign errors in rational functions: ing derivatives like (1-x²)/(x²+1)², students sometimes incorrectly factor the denominator's sign. Remember the denominator squared is always ≥ 0.
- Using test points unwisely: Pick simple numbers inside each interval; avoid endpoints and critical points. If your test value gives zero, choose another.
- 忘记定义域：始终检查原函数在何处有定义。导数在定义域的“空洞”内发生的符号变化不适用。如果 f 在 x = a 处存在垂直渐近线，即便导数符号未变，单调区间也必须在 a 处断开。
- 混淆递增与正导数：f'(x) > 0 当然能保证严格递增，但即便 f'(x) 在某些孤立点等于零（例如 x³ 在 x=0），函数仍可递增。正确的条件是 f'(x) ≥ 0 且不在任何子区间内恒为零。
- 有理函数的符号错误：化简形如 (1-x²)/(x²+1)² 的导数时，有些同学会错误地处理分母的符号。请记住分母平方后始终 ≥ 0。
- 测试点选取不当：在每个区间内选择简单的数值；避开端点和临界点。如果测试值使导数为零，就另选一个。
12. Connecting to Second Derivative and Concavity | 链接二阶导数与凹凸性
While the first derivative reveals where a function increases or decreases, the second derivative f''(x) tells us about concavity. An increasing function can be concave up (f'' > 0) or concave down (f'' < 0). For a complete sketch, IB often asks you to combine monotonicity with concavity to find local maxima/minima and points of inflection. A typical exam prompt might be: "Find the intervals on which f is increasing or decreasing. Hence determine the nature of each stationary point."
一阶导数揭示函数的递增与递减，而二阶导数 f''(x) 告诉我们函数的凹凸性。一个递增函数可以是凹向上的（f'' > 0），也可以是凹向下的（f'' < 0）。为了完整绘制图像，IB经常要求你将单调性与凹凸性结合起来，以求出局部极值点和拐点。典型的考题可能是：“求 f 的递增和递减区间，并由此确定每个驻点的性质。”
13. Summary and Key Takeaways | 总结与核心要点
To master increasing and decreasing functions, always (1) compute f'(x), (2) solve f') = 0 for critical points, (3) draw a sign chart, and (4) label intervals as increasing (f' > 0) or decreasing (f' < 0). Keep in mind that isolated points where f' With practice, monotonicity analysis becomes a fast, reliable tool for calculus problems.
掌握增函数与减函数的关键在于始终要：(1) 计算 f'(x)，(2) 解 f'(x) = 0 求临界点，(3) 绘制符号表，(4) 将区间标记为递增（f' > 0）或递减（f' < 0）。牢记，若导数符号未变，孤立点上的 f' = 0 不会破坏严格单调性。明智地选取测试点，并在任何不连续处将区间断开。通过练习，单调性分析会成为你解决微积分问题快速而可靠的工具。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图