16A Maximum and Minimum Points | 16A 最大值和最小值点

📚 16A Maximum and Minimum Points | 16A 最大值和最小值点

In IB Mathematics, one of the most powerful applications of differential calculus is the ability to locate and classify the highest and lowest points on a graph. These maximum and minimum points, also called extrema, reveal crucial information about the behaviour of a function – from modelling a projectile’s peak height to minimising the cost of production in an optimisation problem. This article explores the concepts, tests and worked examples you need to confidently identify and distinguish between local maxima, local minima, stationary points and points of inflection.

在 IB 数学中，微分学最强大的应用之一就是能够定位并分类图像上的最高点和最低点。这些最大值与最小值点（也称极值点）揭示了函数行为的关键信息——从模拟抛射体的最高高度到优化问题中的生产成本最小化。本文将深入探讨相关概念、判别法和典型例题，帮助你自信地识别并区分局部极大值、局部极小值、驻点和拐点。

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1. What Are Maximum and Minimum Points? | 什么是最大值与最小值点？

A maximum point is a point where a function reaches a peak – the y‑value is greater than the y‑values of all neighbouring points. A minimum point is a valley where the y‑value is lower than its immediate surroundings. Formally, for a function f(x), we say that f(a) is a local maximum if f(a) ≥ f(x) for all x in some open interval around a, and a local minimum if f(a) ≤ f(x) in that interval. The terms ‘local’ (or ‘relative’) are used because these extreme values only need to be the highest or lowest in a small neighbourhood, not necessarily over the entire domain.

最大值点是函数达到峰值的点——其 y 值大于所有邻近点的 y 值。最小值点则是谷底，y 值比周围点都低。严格地说，对于函数 f(x)，如果在 a 附近的某个开区间内对所有 x 都有 f(a) ≥ f(x)，则称 f(a) 为局部极大值；如果 f(a) ≤ f(x)，则称为局部极小值。之所以用“局部”（或“相对”），是因为这些极值只需要在一个小邻域内最高或最低，而不一定在整个定义域上。

When we consider the entire domain of a function, the absolute or global maximum is the largest function value anywhere, and the absolute minimum is the smallest value. Not every function possesses global extrema, but on a closed, bounded interval a continuous function is guaranteed by the Extreme Value Theorem to have both a global maximum and a global minimum.

当考虑函数的整个定义域时，绝对（全局）最大值是函数在所有地方的最大值，绝对最小值则是最小值。并非每个函数都有全局极值，但在一个闭的有界区间上，根据极值定理，连续函数一定同时拥有全局最大值和全局最小值。

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2. Stationary Points and Turning Points | 驻点与转折点

A stationary point occurs where the first derivative of a function equals zero, i.e. f'(x) = 0. Geometrically, the gradient of the tangent at a stationary point is horizontal. Stationary points can be classified as local maxima, local minima or stationary points of inflection. If a stationary point is a local maximum or minimum, we often call it a turning point because the graph changes direction there.

驻点出现在函数的一阶导数等于零的位置，即 f'(x) = 0。从几何上看，驻点处切线的斜率为零（水平）。驻点可以分为局部极大值、局部极小值或驻拐点。如果驻点是局部极大值或极小值，我们通常称其为转折点，因为图像在此处改变方向。

It is important to remember that not every stationary point is a turning point – a stationary point of inflection occurs when f'(x) = 0 but the graph does not change from increasing to decreasing (or vice versa); instead it levels off momentarily and then continues in the same direction. Identifying which type of stationary point we have requires further tests.

务必记住，并非每个驻点都是转折点——当 f'(x) = 0 但图像并未从递增变为递减（或相反），而是瞬间变平然后继续原来方向时，就出现了驻拐点。要判断驻点属于哪一类，我们需要进一步的判别法。

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3. The First Derivative Test | 一阶导数判别法

The first derivative test examines the sign of f'(x) on either side of a stationary point at x = c. If f'(x) changes from positive to negative as x increases through c, then f(c) is a local maximum. If f'(x) changes from negative to positive, f(c) is a local minimum. If the sign does not change, then c is a stationary point of inflection.

一阶导数判别法考察在驻点 x = c 两侧 f'(x) 的符号。如果随着 x 增大经过 c，f'(x) 由正变负，则 f(c) 为局部极大值。如果 f'(x) 由负变正，则 f(c) 为局部极小值。如果符号不变，c 就是一个驻拐点。

This test is especially useful when the second derivative is difficult to compute or equals zero. To apply it, pick a test value just to the left of c and another just to the right, then evaluate the sign of f'(x) at those points. You can record your findings in a simple sign table.

当二阶导数难以计算或等于零时，该判别法特别有用。应用时，在 c 左边和右边各取一个测试值，计算这些点上 f'(x) 的符号，然后将结果记录在一个简单的符号表中。

• Left of c: f'(x) > 0 means the function is increasing; f'(x) < 0 means it is decreasing.

  c 的左侧：f'(x) > 0 表示函数递增；f'(x) < 0 表示函数递减。

• Right of c: f'(x) > 0 means increasing; f'(x) < 0 means decreasing.

  c 的右侧：f'(x) > 0 表示递增；f'(x) < 0 表示递减。

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4. The Second Derivative Test | 二阶导数判别法

The second derivative test provides a quick way to classify a stationary point at x = c, provided that f”(c) exists and is not zero. If f'(c) = 0, we evaluate f”(c):

二阶导数判别法提供了一种快速分类驻点 x = c 的方法，前提是 f”(c) 存在且不为零。如果 f'(c) = 0，我们计算 f”(c)：

• If f”(c) > 0, the graph is concave up at c, and f(c) is a local minimum.

  若 f”(c) > 0，图像在 c 处下凸，f(c) 为局部极小值。

• If f”(c) < 0, the graph is concave down at c, and f(c) is a local maximum.

  若 f”(c) < 0，图像在 c 处上凸，f(c) 为局部极大值。

• If f”(c) = 0, the second derivative test is inconclusive. The point could be a maximum, a minimum or a point of inflection, and you must use the first derivative test instead.

  若 f”(c) = 0，二阶导数判别法无法得出结论。该点可能是极大值、极小值或拐点，此时必须改用一阶导数判别法。

The second derivative test relies on the relationship between concavity and the nature of extrema: a function that is concave up at a critical point must have a local minimum there, and a function that is concave down must have a local maximum.

二阶导数判别法依赖于凹性与极值性质之间的关系：在临界点处下凸的函数必有局部极小值，上凸的函数必有局部极大值。

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5. Classifying Stationary Points: A Summary | 驻点分类总结

The table below summarises the two main tests for classifying stationary points at x = c where f'(c) = 0. You may use whichever method is more efficient for a given function, but always have the first derivative test as a backup when f”(c) = 0.

下表总结了当 f'(c) = 0 时分类驻点的两种主要方法。对于给定的函数，你可以选用更高效的方法，但当 f”(c) = 0 时，一定要用一阶导数判别法作为备用。

Test / 判别法	Local Maximum / 局部极大值	Local Minimum / 局部极小值	Stationary Inflection / 驻拐点
First Derivative / 一阶导数	Sign change + → − / 符号由正变负	Sign change − → + / 符号由负变正	No sign change / 符号不变
Second Derivative / 二阶导数	f”(c) < 0	f”(c) > 0	Inconclusive if f”(c) = 0 / 若 f”(c) = 0 则无法判定

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6. Global (Absolute) vs Local (Relative) Extrema | 全局极值与局部极值

A local extremum is the highest or lowest point within a small neighbourhood, but a global extremum is the highest or lowest value over the entire domain. For instance, the function f(x) = x³ − 3x has a local maximum at x = −1 and a local minimum at x = 1, but it has no global maximum or minimum because its values approach ±∞ as x → ±∞. In contrast, the function f(x) = x² on the domain ℝ has a global minimum at x = 0 but no global maximum.

局部极值是一个小邻域内的最高或最低点，而全局极值是在整个定义域上的最高或最低值。例如，函数 f(x) = x³ − 3x 在 x = −1 处有一个局部极大值，在 x = 1 处有一个局部极小值，但没有全局最大值或最小值，因为当 x → ±∞ 时函数值趋向 ±∞。相比之下，函数 f(x) = x² 在 ℝ 上有全局最小值 x = 0，但没有全局最大值。

When a problem asks for the ‘maximum’ or ‘minimum’ on a closed interval [a, b], you must evaluate the function at all stationary points inside the interval and at the endpoints a and b. The largest of these values is the absolute maximum on [a, b], and the smallest is the absolute minimum. This method is essential for many IB optimisation questions.

当问题要求在闭区间 [a, b] 上求“最大值”或“最小值”时，你必须计算函数在区间内所有驻点以及端点 a 和 b 处的值。这些值中最大的就是 [a, b] 上的绝对最大值，最小的就是绝对最小值。这种方法对许多 IB 优化题至关重要。

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7. Finding Maximum and Minimum Points on a Closed Interval | 在闭区间上求最大值与最小值

To find the absolute maximum and minimum of a continuous function f(x) on a closed interval [a, b], follow these systematic steps:

要求连续函数 f(x) 在闭区间 [a, b] 上的绝对最大值和最小值，请按以下系统步骤操作：

• Step 1: Differentiate to obtain f'(x) and solve f'(x) = 0 to find all stationary points within (a, b).

  第1步：求导得到 f'(x)，解方程 f'(x) = 0，找出 (a, b) 内所有驻点。

• Step 2: Evaluate f(x) at each stationary point.

  第2步：计算每个驻点处的函数值 f(x)。

• Step 3: Evaluate f(a) and f(b).

  第3步：计算 f(a) 和 f(b)。

• Step 4: Compare all values from Steps 2 and 3; the largest is the absolute maximum, the smallest is the absolute minimum.

  第4步：比较步骤2和3中所有的值；最大者为绝对最大值，最小者为绝对最小值。

This technique also works for functions that have points where f'(x) does not exist (cusps, vertical tangents), as long as the function remains continuous. Always include such critical points in your list of candidates.

这一方法同样适用于存在 f'(x) 不存在的点（尖点、垂直切线）的函数，只要函数保持连续。务必将这些临界点也列入候选列表。

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8. Points of Inflection | 拐点

A point of inflection is a point where the concavity of a graph changes. At a non‑stationary point of inflection, f'(x) ≠ 0 but f”(x) = 0 or is undefined, and the second derivative changes sign around that point. At a stationary point of inflection, f'(x) = 0 and the concavity changes; this is a special case where the graph momentarily levels off. The second derivative test alone may not detect a point of inflection – you need to analyse sign changes in f”(x).

拐点是图像的凹性发生改变的点。在非驻拐点处，f'(x) ≠ 0 但 f”(x) = 0 或未定义，且二阶导数在点两侧改变符号。在驻拐点处，f'(x) = 0 且凹性改变；这是图像瞬间走平的特殊情况。仅靠二阶导数判别法可能无法识别拐点——你需要分析 f”(x) 的符号变化。

Points of inflection often accompany maximum‑minimum problems because they help to fully describe the shape of a curve. For example, the cubic f(x) = x³ has a stationary point of inflection at x = 0, where f'(0) = 0 and f”(0) = 0, with the curve changing from concave down to concave up.

拐点常伴随极大极小值问题出现，因为它们有助于完整描述曲线的形状。例如，三次函数 f(x) = x³ 在 x = 0 处有一个驻拐点，满足 f'(0) = 0 且 f”(0) = 0，曲线由上凸变为下凸。

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9. Worked Example 1: Polynomial Function | 示例1：多项式函数

Let us find and classify all stationary points of f(x) = x³ − 3x + 2.

我们来求 f(x) = x³ − 3x + 2 的所有驻点并进行分类。

f'(x) = 3x² − 3 = 0 ⟹ 3(x² − 1) = 0 ⟹ x = −1 or x = 1

Now compute the second derivative: f”(x) = 6x.

现在计算二阶导数：f”(x) = 6x。

• At x = −1: f”(−1) = −6 < 0, so f(−1) = (−1)³ − 3(−1) + 2 = 4 is a local maximum.

  当 x = −1 时：f”(−1) = −6 < 0，因此 f(−1) = 4 为局部极大值。

• At x = 1: f”(1) = 6 > 0, so f(1) = (1)³ − 3(1) + 2 = 0 is a local minimum.

  当 x = 1 时：f”(1) = 6 > 0，因此 f(1) = 0 为局部极小值。

The stationary points are (−1, 4) [maximum] and (1, 0) [minimum]. A sign analysis of f'(x) confirms that f'(x) changes from positive to negative at x = −1 and from negative to positive at x = 1.

驻点为 (−1, 4) [极大值] 和 (1, 0) [极小值]。对 f'(x) 的符号分析也证实，在 x = −1 处 f'(x) 由正变负，在 x = 1 处由负变正。

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10. Worked Example 2: Rational Function | 示例2：有理函数

Consider the function g(x) = x + 4/x for x > 0. This is a typical rational function where locating the minimum point is useful for optimisation.

考虑函数 g(x) = x + 4/x (x > 0)。这是一个典型的有理函数，求其最小值点对优化问题很有用。

g'(x) = 1 − 4/x² = 0 ⟹ x² = 4 ⟹ x = 2 (since x > 0)

g”(x) = 8/x³

At x = 2, g”(2) = 8/8 = 1 > 0, indicating a local minimum. The minimum value is g(2) = 2 + 4/2 = 4. Therefore, the point (2, 4) is a local minimum. Because g(x) → ∞ as x → 0⁺ or x → ∞, this is also the global minimum on (0, ∞).

在 x = 2 处，g”(2) = 1 > 0，表明这是一个局部极小值。极小值为 g(2) = 4。因此点 (2, 4) 是局部极小值。由于当 x → 0⁺ 或 x → ∞ 时 g(x) → ∞，该点也是 (0, ∞) 上的全局最小值。

The first derivative test would likewise confirm that g'(x) < 0 for 0 < x < 2 and g'(x) > 0 for x > 2, so the function decreases to the minimum and then increases.

一阶导数判别法同样会证实，当 0 < x < 2 时 g'(x) < 0，当 x > 2 时 g'(x) > 0，因此函数先递减至最小值然后递增。

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11. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

Maximum and minimum problems can trip up even confident students. Here are some classic errors and tips to steer clear of them:

即使是自信的学生也可能在极大极小值问题上栽跟头。以下是一些经典错误及避免建议：

• Forgetting to check the nature of stationary points: Simply setting f'(x) = 0 is not enough; you must classify the point using a derivative test. Memorise both tests and practise them.

  忘记验证驻点性质：仅仅令 f'(x) = 0 是不够的；你必须用导数判别法对驻点进行分类。把两种方法都记牢并多加练习。

• Ignoring endpoints in closed‑interval problems: The global extremum may occur at an endpoint, not at a stationary point. Always evaluate f(a) and f(b).

  在闭区间问题中忽略端点：全局极值可能出现在端点而非驻点。务必计算 f(a) 和 f(b)。

• Misinterpreting f”(x) = 0: A zero second derivative means the test fails; it does not automatically indicate a point of inflection. Use the first derivative test or check the sign change of f”(x).

  误解 f”(x) = 0：二阶导数为零意味着判别法失效；它并不自动表明是拐点。应使用一阶导数判别法或检查 f”(x) 的符号变化。

• Confusing local and global extrema: A local maximum may be lower than a local minimum elsewhere in the domain. Always define the interval or domain when stating ‘maximum’ or ‘minimum’.

  混淆局部与全局极值：局部极大值可能比定义域中别处的局部极小值还要小。在描述“最大值”或“最小值”时，务必明确区间或定义域。

• Algebraic slips when solving f'(x) = 0: Carefully factorise, use the quadratic formula when needed, and double‑check your solutions. One sign error can move the whole stationary point.

  解 f'(x) = 0 时的代数错误：仔细因式分解，需要时使用二次公式，并反复检查你的解。一个符号错误就会使整个驻点移位。

• Not simplifying the derivative before solving: This can lead to unnecessary complexity. Cancel common factors and simplify rational expressions where possible.

  解方程前不化简导数：这会导致不必要的复杂性。尽量消去公因子并化简有理式。

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12. Conclusion | 总结

Maximum and minimum points are a cornerstone of calculus in the IB Mathematics syllabus. By mastering the concepts of stationary points, the first and second derivative tests, and the distinction between local and global extrema, you will be well equipped to tackle a wide variety of problems. The key is to be systematic: differentiate, find critical values, test their nature, and always consider the context – whether you are analysing an unrestricted function or working on a closed interval. With practice, classifying extrema becomes a swift, reliable skill that opens the door to deeper mathematical modelling and optimisation tasks.

最大值与最小值点是 IB 数学课程中微积分的基础。通过掌握驻点概念、一阶和二阶导数判别法以及局部与全局极值的区别，你将能从容应对各种各样的问题。关键是系统化：求导、找出临界值、检验其性质，并始终考虑上下文——无论是分析无约束的函数，还是在闭区间上求解。经过练习，极值分类将成为一项迅速、可靠的技能，为你打开更深层次的数学建模与优化任务的大门。

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