IB Mathematics: Methods for Determining Increasing and Decreasing Functions | IB数学：函数增减性的判断方法

📚 IB Mathematics: Methods for Determining Increasing and Decreasing Functions | IB数学：函数增减性的判断方法

In IB Mathematics, understanding whether a function is increasing or decreasing is essential for analysing its behaviour, sketching graphs, and solving optimisation problems. This article presents the main methods used to determine intervals of increase and decrease.

在IB数学中，理解函数是递增还是递减对于分析其性质、绘制图像以及解决最优化问题至关重要。本文介绍判断函数增减区间的主要方法。

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1. Definition of Increasing and Decreasing Functions | 函数增减性的定义

A function f is said to be strictly increasing on an interval I if, for any x₁ < x₂ in I, we have f(x₁) < f(x₂). Similarly, f is strictly decreasing on I if f(x₁) > f(x₂) whenever x₁ < x₂.

如果对于区间I中的任意x₁ < x₂，都有f(x₁) < f(x₂)，则称函数f在区间I上严格递增。类似地，若x₁ < x₂时总有f(x₁) > f(x₂)，则称f在I上严格递减。

Some texts also use the terms non-decreasing and non-increasing when the inequalities are not strict, i.e. f(x₁) ≤ f(x₂) or f(x₁) ≥ f(x₂).

一些教材还使用“非递减”和“非递增”的术语，表示不等式不严格，即f(x₁) ≤ f(x₂)或f(x₁) ≥ f(x₂)。

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2. Using the Definition to Verify Monotonicity | 用定义判定单调性

The definition can be applied directly. For a given function, pick two arbitrary points x₁ < x₂ in the interval and examine the sign of f(x₂) − f(x₁).

可以直接应用定义。对于给定函数，在区间内任取两点x₁ < x₂，考察f(x₂) − f(x₁)的符号。

• If f(x₂) − f(x₁) > 0 for all such pairs, the function is increasing.

  如果对所有这样的点对都有f(x₂) − f(x₁) > 0，则函数递增。

• If f(x₂) − f(x₁) < 0 for all such pairs, the function is decreasing.

  如果对所有这样的点对都有f(x₂) − f(x₁) < 0，则函数递减。

This method is especially useful for functions that are not differentiable, such as piecewise-defined functions.

此方法尤其适用于不可导的函数，例如分段函数。

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3. The Derivative Test | 导数判定法

If a function f is differentiable on an open interval (a, b), then the sign of its derivative f′(x) determines monotonicity:

如果函数f在开区间(a, b)内可导，则其导数f′(x)的符号决定单调性：

• If f′(x) > 0 for all x in (a, b), then f is strictly increasing on [a, b].

  如果对(a, b)内所有x都有f′(x) > 0，则f在[a, b]上严格递增。

• If f′(x) < 0 for all x in (a, b), then f is strictly decreasing on [a, b].

  如果对(a, b)内所有x都有f′(x) < 0，则f在[a, b]上严格递减。

Note that the derivative may equal zero at isolated points without affecting the overall monotonicity.

注意导数值在个别点处可以等于零，这并不影响整体单调性。

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4. Critical Points and Sign Intervals | 临界点与符号区间

To find intervals of increase and decrease, we locate critical points, where f′(x) = 0 or f′(x) does not exist. These points split the domain into intervals.

为了找到增减区间，我们需确定临界点，即f′(x) = 0或f′(x)不存在的点。这些点将定义域划分成若干区间。

Within each interval, we choose a test value and evaluate the sign of f′ to decide whether the function is increasing or decreasing there.

在每个区间内，我们选取一个测试值并判断f′的符号，以确定函数在该区间是递增还是递减。

Example: f(x) = x³ − 3x + 2 → f′(x) = 3x² − 3 = 3(x − 1)(x + 1)

The critical points are x = −1 and x = 1. They divide the real line into (−∞, −1), (−1, 1), (1, ∞).

临界点为x = −1和x = 1。它们将实数轴分为(−∞, −1)、(−1, 1)、(1, ∞)三个区间。

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5. Practical Steps for the Derivative Method | 导数法的实用步骤

Here is a systematic procedure for determining monotonicity intervals using derivatives.

以下是使用导数判断单调区间的系统步骤。

1. Find the domain of f.

   求出函数f的定义域。

2. Compute f′(x).

   计算f′(x)。

3. Solve f′(x) = 0 and find where f′(x) is undefined.

   解方程f′(x) = 0，并找出f′(x)无定义的点。

4. Use these critical numbers to divide the domain into test intervals.

   用这些临界点将定义域划分为测试区间。

5. Pick a test point in each interval and determine the sign of f′.

   在每个区间内选取测试点，确定f′的符号。

6. Conclude which intervals are increasing (f′ > 0) and which are decreasing (f′ < 0).

   根据f′ > 0递增、f′ < 0递减得出相应区间。

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6. Increasing and Decreasing on an Interval vs at a Point | 区间增减与点处增减

Strictly speaking, monotonicity is a property of a function on an interval, not at a single point. However, the sign of the derivative at a point describes the local behaviour of the tangent line.

严格来说，单调性是函数在区间上的性质，而不是单点的性质。然而，某点处导数的符号描述了切线的局部行为。

If f′(a) > 0, the function is increasing in a small neighbourhood around a. This does not guarantee global increase over a larger interval.

如果f′(a) > 0，则函数在a附近的小邻域内递增。但这并不保证在更大区间上全局递增。

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7. Common Pitfalls | 常见错误

Students often make mistakes when applying these tests. Here are the most frequent errors to avoid.

学生在应用这些判定方法时常犯错误。以下是最常见的错误，需要避免。

• Confusing the domain of f with the interval where f′ keeps a constant sign.

  混淆函数定义域与f′保持恒定符号的区间。

• Ignoring points where f′ is undefined, such as corners or vertical tangents.

  忽略f′不存在的点，如尖点或垂直切线点。

• Assuming f′(x) = 0 always indicates a change of monotonicity.

  认为f′(x) = 0必然导致单调性改变。

• Forgetting to consider discontinuity: a function can be increasing on separate intervals yet drop at a vertical asymptote.

  忘记考虑不连续性：函数在各自区间上可以递增，但在垂直渐近线处可能发生突变下降。

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8. Worked Example | 例题详解

Let us apply the method to the function f(x) = x³ − 3x + 2 on its entire domain.

让我们在完整定义域上对函数f(x) = x³ − 3x + 2应用上述方法。

Step 1: Domain is all real numbers. Step 2: f′(x) = 3x² − 3.

第一步：定义域为全体实数

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