Increasing and Decreasing Functions | 递增与递减函数

📚 Increasing and Decreasing Functions | 递增与递减函数

In pure mathematics, understanding how a function behaves as its input changes is fundamental. This article focuses on the concepts of increasing and decreasing functions, their connection to the derivative, and how to determine these properties rigorously for Edexcel A-Level Mathematics.

在纯数学中，理解函数随自变量变化而如何变化是基础。本文聚焦于递增与递减函数的概念、它们与导数的联系，以及如何为 Edexcel A-Level 数学严格地判断这些性质。

1. What Are Increasing and Decreasing Functions? | 什么是递增与递减函数？

A function \(f(x)\) is said to be increasing on an interval if, for any two values \(x_1\) and \(x_2\) in that interval with \(x_1 < x_2\), we have \(f(x_1) \le f(x_2)\). If the inequality is strict, i.e. \(f(x_1) < f(x_2)\), the function is strictly increasing on that interval.

若函数 \(f(x)\) 在某个区间内，对于任意两个值 \(x_1\) 和 \(x_2\)（其中 \(x_1 < x_2\)），均有 \(f(x_1) \le f(x_2)\)，则称该函数在该区间上递增。若不等式严格成立，即 \(f(x_1) < f(x_2)\)，则称函数在该区间上严格递增。

Similarly, a function is decreasing on an interval if \(x_1 < x_2\) implies \(f(x_1) \ge f(x_2)\). A strict inequality \(f(x_1) > f(x_2)\) gives a strictly decreasing function.

类似地，若 \(x_1 < x_2\) 蕴含 \(f(x_1) \ge f(x_2)\)，则函数在该区间上递减。严格不等式 \(f(x_1) > f(x_2)\) 则对应严格递减函数。

2. The Role of the Derivative | 导数的作用

For differentiable functions, the sign of the derivative \(f'(x)\) provides a direct test. If \(f'(x) > 0\) for all \(x\) in an interval, then the function is strictly increasing on that interval. If \(f'(x) < 0\), the function is strictly decreasing.

对于可微函数，导数 \(f'(x)\) 的符号提供了直接判定方法。若在某个区间内对所有 \(x\) 都有 \(f'(x) > 0\)，则函数在该区间上严格递增；若 \(f'(x) < 0\)，则函数严格递减。

This result follows from the Mean Value Theorem, but for A-Level purposes it is sufficient to remember that the derivative measures the instantaneous rate of change. A positive slope means the graph rises as \(x\) increases; a negative slope means the graph falls.

这一结果可由中值定理推出，但在 A-Level 阶段只需记住导数衡量瞬时变化率即可：斜率为正意味着图形随 \(x\) 增大而上升；斜率为负意味着图形下降。

3. Increasing Functions on an Interval | 区间上的递增函数

Consider the function \(f(x) = x^3 + 2x\). Its derivative is \(f'(x) = 3x^2 + 2\). Since \(3x^2 \ge 0\) for all real \(x\), we have \(f'(x) \ge 2 > 0\) everywhere. Hence \(f(x)\) is strictly increasing on the entire real line \(\mathbb{R}\).

考虑函数 \(f(x) = x^3 + 2x\)。其导数为 \(f'(x) = 3x^2 + 2\)。因为对所有实数 \(x\) 都有 \(3x^2 \ge 0\)，所以 \(f'(x) \ge 2 > 0\) 处处成立。因此 \(f(x)\) 在整个实数集 \(\mathbb{R}\) 上严格递增。

Notice that an increasing function does not necessarily have a positive derivative everywhere. For example, \(f(x) = x^3\) is strictly increasing, but \(f'(0) = 0\). The derivative can be zero at isolated points without destroying the overall increasing property.

注意，递增函数并不一定处处都有正导数。例如，\(f(x) = x^3\) 是严格递增的，但 \(f'(0) = 0\)。导数可以在个别点为零而不影响整体的递增性质。

4. Decreasing Functions on an Interval | 区间上的递减函数

Let \(g(x) = -x^2 + 4x – 1\). Its derivative is \(g'(x) = -2x + 4\). For \(x > 2\), \(g'(x) < 0\), so the function is decreasing on the interval \((2, \infty)\). For \(x < 2\), \(g'(x) > 0\), so it is increasing on \((-\infty, 2)\).

设 \(g(x) = -x^2 + 4x – 1\)。其导数为 \(g'(x) = -2x + 4\)。当 \(x > 2\) 时，\(g'(x) < 0\)，因此函数在区间 \((2, \infty)\) 上递减；当 \(x < 2\) 时，\(g'(x) > 0\)，因此在 \((-\infty, 2)\) 上递增。

This illustrates that a function can be increasing on one interval and decreasing on another. The point where the derivative changes sign is often a local maximum or minimum.

这表明一个函数可以在一个区间递增而在另一个区间递减。导数改变符号的点通常是局部最大值或最小值。

5. Critical Points and Stationary Points | 临界点与驻点

A stationary point (or turning point) occurs where \(f'(x) = 0\). These points are important because they separate intervals where a function increases from those where it decreases. A critical point is any point where the derivative is zero or undefined.

驻点（或转折点）出现在 \(f'(x) = 0\) 处。这些点很重要，因为它们将函数递增的区间与递减的区间分开。临界点是导数为零或不存在的点。

For a polynomial or other differentiable function, finding where \(f'(x) = 0\) gives the locations of possible local maxima, local minima, or points of inflection that are horizontal.

对于多项式或其他可微函数，求出 \(f'(x) = 0\) 的位置可以得到可能的局部最大值、局部最小值或水平拐点。

6. Testing with the First Derivative | 一阶导数判定法

To determine whether a function is increasing or decreasing on either side of a critical point, we can use a sign table. Pick test values in the intervals between critical points and evaluate the sign of \(f'(x)\).

为了确定临界点两侧函数的递增或递减情况，可以使用符号表。选取临界点之间各区间的测试值，判断 \(f'(x)\) 的符号。

Consider \(f(x) = x^3 – 3x\). Then \(f'(x) = 3x^2 – 3 = 3(x – 1)(x + 1)\). The critical points are \(x = -1\) and \(x = 1\). For \(x < -1\), \(f'(x) > 0\); for \(-1 < x < 1\), \(f'(x) < 0\); for \(x > 1\), \(f'(x) > 0\). Hence \(f\) is increasing on \((-\infty, -1)\), decreasing on \((-1, 1)\), and increasing again on \((1, \infty)\).

考虑 \(f(x) = x^3 – 3x\)。则 \(f'(x) = 3x^2 – 3 = 3(x – 1)(x + 1)\)。临界点为 \(x = -1\) 和 \(x = 1\)。当 \(x < -1\) 时，\(f'(x) > 0\)；当 \(-1 < x < 1\) 时，\(f'(x) < 0\)；当 \(x > 1\) 时，\(f'(x) > 0\)。因此 \(f\) 在 \((-\infty, -1)\) 上递增，在 \((-1, 1)\) 上递减，又在 \((1, \infty)\) 上递增。

7. Graphical Interpretation | 图形解释

On a graph, an increasing function rises from left to right, while a decreasing function falls from left to right. The derivative at a point is the slope of the tangent line, so a positive slope corresponds to an upward trend, and a negative slope corresponds to a downward trend.

在图形上，递增函数从左向右上升，递减函数从左向右下降。函数在某一点的导数是切线的斜率，因此正斜率对应上升趋势，负斜率对应下降趋势。

At a local maximum, the function changes from increasing to decreasing, so the derivative changes from positive to negative. At a local minimum, the derivative changes from negative to positive.

在局部最大值处，函数由递增变为递减，因此导数由正变负；在局部最小值处，导数由负变正。

8. Worked Example | 例题解析

Find the intervals on which \(f(x) = 2x^3 – 9x^2 + 12x – 5\) is increasing or decreasing.

求函数 \(f(x) = 2x^3 – 9x^2 + 12x – 5\) 的递增与递减区间。

First, differentiate: \(f'(x) = 6x^2 – 18x + 12 = 6(x^2 – 3x + 2) = 6(x – 1)(x – 2)\).

首先求导：\(f'(x) = 6x^2 – 18x + 12 = 6(x^2 – 3x + 2) = 6(x – 1)(x – 2)\)。

Set \(f'(x) = 0\), giving \(x = 1\) and \(x = 2\). Test the sign in three intervals: \(x < 1\), \(1 < x < 2\), \(x > 2\). Choose \(x = 0\): \(f'(0) = 12 > 0\). Choose \(x = 1.5\): \(f'(1.5) = 6(-0.5)(-0.5) = 1.5 > 0\). Wait, that would be positive, but let’s recalculate: \(6(1.5 – 1)(1.5 – 2) = 6(0.5)(-0.5) = -1.5 < 0\). Correction: \(f'(1.5) = -1.5\). Choose \(x = 3\): \(f'(3) = 6(2)(1) = 12 > 0\).

令 \(f'(x) = 0\)，得 \(x = 1\) 和 \(x = 2\)。在三个区间内测试符号：\(x < 1\)、\(1 < x < 2\)、\(x > 2\)。取 \(x = 0\)：\(f'(0) = 12 > 0\)。取 \(x = 1.5\)：\(f'(1.5) = 6(0.5)(-0.5) = -1.5 < 0\)。取 \(x = 3\)：\(f'(3) = 6(2)(1) = 12 > 0\)。

Thus the function is increasing on \((-\infty, 1)\) and \((2, \infty)\), and decreasing on \((1, 2)\).

因此函数在 \((-\infty, 1)\) 和 \((2, \infty)\) 上递增，在 \((1, 2)\) 上递减。

9. Common Pitfalls | 常见误区

Assuming that \(f'(x) > 0\) at a single point proves the function is increasing on an interval. You must check all points in the interval, or at least know that the derivative is positive throughout.

误以为仅凭某一点 \(f'(x) > 0\) 就能证明函数在区间上递增。必须检查区间内所有点，或者知道导数在整个区间上为正。
Confusing strictly increasing with non-decreasing. A function with \(f'(x) \ge 0\) may have horizontal segments; it is non-decreasing but not always strictly increasing.

混淆严格递增与非递减。若 \(f'(x) \ge 0\)，函数可能有水平段；它是非递减但不一定是严格递增。
Forgetting that a function may be increasing even if the derivative is zero at isolated points, as in \(f(x) = x^3\).

忘记函数即使在个别点导数为零仍可能是递增的，例如 \(f(x) = x^3\)。

10. Summary | 总结

To analyse increasing and decreasing behaviour, compute the derivative, find the critical points, and test the sign of the derivative in each interval. A positive derivative indicates an increasing function, and a negative derivative indicates a decreasing function.

要分析递增与递减行为，需计算导数、找到临界点，并在每个区间内测试导数的符号。正导数表示函数递增，负导数表示函数递减。

Situation	Derivative sign	Conclusion
Increasing	\(f'(x) > 0\)	Function rises
Decreasing	\(f'(x) < 0\)	Function falls
Stationary	\(f'(x) = 0\)	Potential turning point

Mastering this skill is essential for curve sketching, optimisation problems, and interpreting rates of change in applied contexts.

掌握这一技能对于曲线绘图、优化问题以及在实际情境中解释变化率至关重要。

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Increasing and Decreasing Functions | 递增与递减函数