Maxima and Minima Points of a Function | 函数的最大值与最小值点

📚 Maxima and Minima Points of a Function | 函数的最大值与最小值点

In IB Mathematics, understanding maxima and minima is essential for analysing the behaviour of functions. These points help us interpret graphs, solve optimisation problems, and model real-world situations, from maximising profit to minimising material used in manufacturing.

在IB数学中，理解最大值与最小值是分析函数行为的关键。这些点帮助我们解读图像、解决优化问题，并模拟现实世界中的情境——从最大化利润到最小化生产中的材料用量。

1. Definition of Extrema | 极值的定义

Extrema of a function are its maximum and minimum values. A local maximum occurs at a point where the function value is greater than or equal to all nearby values, while a local minimum occurs where it is lower than or equal to all nearby values.

函数的极值指其最大值与最小值。局部最大值出现在函数值大于或等于所有邻近值之处，而局部最小值出现在函数值小于或等于所有邻近值之处。

A global maximum is the largest value of the entire function, and a global minimum is the smallest. Local extrema are also called relative extrema, because they are only extreme relative to a small neighbourhood.

全局最大值是整个函数的最大取值，全局最小值则是最小取值。局部极值也称为相对极值，因为它们仅在某个小邻域内是极端的。

Local maximum: f(c) ≥ f(x) for all x near c.
Local minimum: f(c) ≤ f(x) for all x near c.
Global maximum: f(c) ≥ f(x) for all x in the domain.
Global minimum: f(c) ≤ f(x) for all x in the domain.

局部最大值：对c附近所有x，满足 f(c) ≥ f(x)。
局部最小值：对c附近所有x，满足 f(c) ≤ f(x)。
全局最大值：对定义域内所有x，满足 f(c) ≥ f(x)。
全局最小值：对定义域内所有x，满足 f(c) ≤ f(x)。

2. Stationary Points and Critical Points | 驻点与临界点

A stationary point is where the derivative equals zero, i.e. f'(x) = 0. These points are horizontal on the graph and are prime candidates for local extrema.

驻点是导数等于零的点，即 f'(x) = 0。这些点在图像上呈水平状，是局部极值的首要候选者。

Critical points include all points where f'(x) = 0, as well as points where f'(x) does not exist, provided the function is continuous at those points. In IB problems, we often examine both types.

临界点包括所有 f'(x) = 0 的点，以及 f'(x) 不存在的点（前提是函数在这些点连续）。在IB题目中，我们通常同时考察这两类点。

Type of point	Condition	Example
Stationary point	f'(x) = 0	f(x) = x² at x = 0
Singular point	f'(x) undefined	f(x) = \|x\| at x = 0

点的类型	条件	例子
驻点	f'(x) = 0	f(x) = x² 在 x = 0 处
奇异点	f'(x) 无定义	f(x) = \|x\| 在 x = 0 处

3. First Derivative Test | 一阶导数测试法

The first derivative test examines the sign change of f'(x) around a critical point. If f'(x) changes from positive to negative, the point is a local maximum; from negative to positive, a local minimum; if no change, it is a point of inflection.

一阶导数测试法考察临界点附近 f'(x) 的符号变化。若 f'(x) 由正变负，该点为局部最大值；由负变正，则为局部最小值；若无变化，则为拐点。

This test is reliable for all differentiable functions and is often the first method taught in IB. It requires evaluating f'(x) just to the left and right of each critical point.

该测试对一切可导函数都可靠，常是IB课程中首先教授的方法。它需要评估每个临界点左右两侧的 f'(x) 值。

Sign of f'(x) | f'(x) 的符号

Positive → Negative : Local maximum | 局部最大值
Negative → Positive : Local minimum | 局部最小值
No change : Point of inflection | 拐点

4. Second Derivative Test | 二阶导数测试法

The second derivative test is a faster alternative. At a stationary point where f'(x) = 0, if f”(x) > 0, the point is a local minimum; if f”(x) < 0, a local maximum.

二阶导数测试法是一种更快捷的替代方法。在 f'(x) = 0 的驻点上，若 f”(x) > 0，该点为局部最小值；若 f”(x) < 0，则为局部最大值。

However, if f”(x) = 0, the test is inconclusive. In that case, we must revert to the first derivative test or examine higher-order derivatives to classify the point.

然而，若 f”(x) = 0，该测试无效。此时，我们必须回到一阶导数测试法，或考察更高阶导数来判别该点。

f”(x)	Conclusion
f”(x) > 0	Local minimum \| 局部最小值
f”(x) < 0	Local maximum \| 局部最大值
f”(x) = 0	Inconclusive \| 无法确定

5. Extrema on a Closed Interval | 闭区间上的最值

On a closed interval [a, b], the global maximum and minimum must occur either at critical points in the interior or at the endpoints a and b. We evaluate all candidates and compare their values.

在闭区间 [a, b] 上，全局最大值和最小值必然出现在内部临界点或端点 a 和 b 处。我们需要评估所有候选点并比较它们的函数值。

The procedure is straightforward: find all critical points inside the interval, compute f at each critical point and at both endpoints, then select the largest and smallest values.

步骤很直接：找出区间内所有临界点，计算每个临界点及两个端点处的 f 值，然后选取最大和最小值。

Steps for closed interval | 闭区间求解步骤

Find f'(x) and solve f'(x) = 0 within (a, b).
Evaluate f at each critical point, plus f(a) and f(b).
Largest value = global max; smallest = global min.

求 f'(x) 并解 f'(x) = 0 于 (a, b) 内。
计算每个临界点的 f 值，再加 f(a) 和 f(b)。
最大值为全局最大；最小值为全局最小。

6. Optimisation Problems | 优化问题

Optimisation is a key application of extrema. We build a function that models a quantity to be maximised or minimised, then use differentiation to find its extrema over a reasonable domain.

优化是极值的重要应用。我们构造一个函数来模拟需要最大化或最小化的量，然后通过微分在合理定义域上求其极值。

For example, given a fixed perimeter, the rectangle with maximum area is a square. We write area = xy, use the perimeter constraint to eliminate one variable, differentiate, and solve.

例如，在固定周长下，面积最大的矩形是正方形。我们写出面积 = xy，利用周长约束消去一个变量，求导并求解。

Example | 示例

Maximise A = x(20 – x) for a rectangle with perimeter 40. Differentiate: A’ = 20 – 2x = 0 → x = 10. Then y = 10, giving a square with maximised area 100.

在周长40条件下，最大化 A = x(20 – x)。求导得 A’ = 20 – 2x = 0 → x = 10。于是 y = 10，即最大面积为100的正方形。

7. Points of Inflection and Their Relation | 拐点及其关系

A point of inflection is where the concavity changes, i.e. f”(x) = 0 and changes sign. Not all points with f”(x) = 0 are inflection points; we must check concavity on both sides.

拐点是函数凹凸性发生改变的点，即 f”(x) = 0 且符号改变。并非所有 f”(x) = 0 的点都是拐点，我们必须检查两侧的凹凸性。

Inflection points are not extrema, but they often appear in exam questions alongside maxima and minima. Understanding them helps us sketch accurate graphs and avoid misclassification.

拐点不是极值点，但它们常与最大最小值一同出现在考试题中。理解拐点有助于我们绘制准确图像，避免误判。

Property	Local max	Local min	Inflection
f'(x)	0	0	may be 0
f”(x)	< 0	> 0	0, sign change

性质	局部最大值	局部最小值	拐点
f'(x)	0	0	可能为0
f”(x)	< 0	> 0	0，且变号

8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

One frequent error is assuming that every point with f'(x) = 0 is an extremum. A counterexample is f(x) = x³, where f'(0) = 0 but the point is an inflection, not a maximum or minimum.

一个常见错误是认为所有 f'(x) = 0 的点都是极值点。反例是 f(x) = x³，虽然 f'(0) = 0，但该点是拐点，而非最大或最小值。

Another mistake occurs when ignoring endpoints on closed intervals, or when applying the second derivative test without verifying f'(x) = 0 first. Always list all candidates and justify your conclusions.

另一个错误是在闭区间上忽略端点，或未先验证 f'(x) = 0 就直接使用二阶导数测试。务必列出所有候选点并论证你的结论。

Do not confuse local and global extrema.
Check that f'(x) changes sign, not just equals zero.
Always include endpoints when solving on [a, b].

切勿混淆局部极值与全局极值。
检查 f'(x) 变号，而不仅仅是等于0。
在 [a, b] 上求解时，务必包含端点。

9. Worked Example: Complete Solution | 完整例题解析

Find the extrema of f(x) = 2x³ – 9x² + 12x + 1 on the interval [0, 3]. First compute f'(x) = 6x² – 18x + 12 = 6(x – 1)(x – 2). Critical points are x = 1 and x = 2.

求函数 f(x) = 2x³ – 9x² + 12x + 1 在区间 [0, 3] 上的极值。首先计算 f'(x) = 6x² – 18x + 12 = 6(x – 1)(x – 2)。临界点为 x = 1 和 x = 2。

Now compute f(0) = 1, f(1) = 6, f(2) = 5, f(3) = 10. Therefore, the global maximum is 10 at x = 3, and the global minimum is 1 at x = 0. Locally, (1, 6) is a maximum and (2, 5) is a minimum, as verified by f”(x) = 12x – 18: f”(1) = -6 < 0, f”(2) = 6 > 0.

接着计算 f(0) = 1，f(1) = 6，f(2) = 5，f(3) = 10。因此，全局最大值为10（x = 3处），全局最小值为1（x = 0处）。局部而言，(1, 6) 为最大值，(2, 5) 为最小值，这可由 f”(x) = 12x – 18 验证：f”(1) = -6 < 0，f”(2) = 6 > 0。

Summary table | 汇总表

x	0	1	2	3
f(x)	1	6	5	10
Conclusion	Global min	Local max	Local min	Global max

x	0	1	2	3
f(x)	1	6	5	10
结论	全局最小值	局部最大值	局部最小值	全局最大值

10. Practice Questions | 练习题目

Attempt these three typical IB-style questions. Solutions follow in the next section, but try solving them independently first.

尝试以下三道典型IB风格题目。解答将在下一节给出，但请先独立求解。

Question 1: Find the local extrema of f(x) = x⁴ – 4x³ + 4x². Question 2: Determine the global max and min of f(x) = sin(x) + cos(x) on [0, π].

问题1：求 f(x) = x⁴ – 4x³ + 4x² 的局部极值。问题2：确定 f(x) = sin(x) + cos(x) 在 [0, π] 上的全局最大和最小值。

Question 3: A farmer wants to fence a rectangular field with 100 m of fencing. What dimensions give the maximum area?

问题3：一位农民用100米篱笆围矩形田地，什么尺寸使面积最大？

11. Quick Solutions and Explanation | 快速解答与讲解

Question 1: f'(x) = 4x³ – 12x² + 8x = 4x(x – 1)(x – 2). Critical points x = 0, 1, 2. Using f”(x) = 12x² – 24x + 8: at x = 0, f” = 8 > 0 (minimum); at x = 1, f” = -4 < 0 (maximum); at x = 2, f” = 8 > 0 (minimum).

问题1：f'(x) = 4x³ – 12x² + 8x = 4x(x – 1)(x – 2)。临界点 x = 0, 1, 2。用 f”(x) = 12x² – 24x + 8：在 x = 0 处，f” = 8 > 0（最小值）；在 x = 1 处，f” = -4 < 0（最大值）；在 x = 2 处，f” = 8 > 0（最小值）。

Question 2: f'(x) = cos(x) – sin(x) = 0 → x = π/4. Evaluate f(0) = 1, f(π/4) = √2 ≈ 1.414, f(π) = -1. So global max is √2 at π/4, global min is -1 at π.

问题2：f'(x) = cos(x) – sin(x) = 0 → x = π/4。计算 f(0) = 1，f(π/4) = √2 ≈ 1.414，f(π) = -1。所以全局最大值为 √2（在 π/4 处），全局最小值为 -1（在 π 处）。

Question 3: Let length = x, width = (100 – 2x)/2 = 50 – x. Area A = x(50 – x). A’ = 50 – 2x = 0 → x = 25, so width = 25. The maximum area is 625 m², a square.

问题3：设长为 x，宽为 (100 – 2x)/2 = 50 – x。面积 A = x(50 – x)。A’ = 50 – 2x = 0 → x = 25，故宽 = 25。最大面积为625平方米，即正方形。

12. Summary and Key Takeaways | 总结与核心要点

Maxima and minima are found by combining derivative analysis with critical-point reasoning. Always classify points carefully using the first or second derivative test, and never forget endpoints in closed intervals.

最大与最小值的求解需要结合导数分析与临界点推理。务必使用一阶或二阶导数测试仔细分类，且绝不要在闭区间上忘记端点。

Practicing with a variety of polynomial, trigonometric, and optimisation problems will solidify your understanding. Keep a clear checklist in mind: find f’, solve f’ = 0, test each point, compare all candidates, and state your answer clearly.

通过练习各类多项式、三角函数和优化问题，你将巩固理解。牢记一条清晰的清单：求 f’，解 f’ = 0，测试每个点，比较所有候选值，并清晰地陈述答案。

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Maxima and Minima Points of a Function | 函数的最大值与最小值点