📚 Maxima and Minima of Functions | 函数的极大值与极小值
In IB Mathematics Analysis & Approaches (AA) and Applications & Interpretation (AI), the study of maxima and minima is a central pillar of differential calculus. From maximising profit in a business model to minimising the material used in a container, the ability to locate where a function attains its greatest or smallest value links abstract theory with practical decision-making. This article provides a systematic, syllabus-aligned review of how to find and classify extrema, with worked examples and exam-style pitfalls clearly explained.
在IB数学”分析与方法”（AA）与”应用与解释”（AI）课程中，极大值与极小值的研究是微分学的核心支柱。从企业模型中的利润最大化到容器用料的最小化，寻找函数取得最大值或最小值的点，将抽象理论与现实决策紧密相连。本文围绕考纲要求，系统讲解如何求取和判定极值，并配有典型例题与考试常见陷阱的细致分析。
1. What Are Maxima and Minima? | 什么是极大值与极小值？
In differential calculus, a function f(x) has a local maximum at x = a if f(a) ≥ f(x) for every x sufficiently close to a. Similarly, it has a local minimum at x = a if f(a) ≤ f(x) for every x sufficiently close to a. The word “local” indicates that the comparison is restricted to a small neighbourhood around a, not the entire domain.
在微分学中，若对a附近足够近的所有x都满足f(a) ≥ f(x)，则称函数f(x)在x = a处取得局部极大值；若满足f(a) ≤ f(x)，则称函数在x = a处取得局部极小值。”局部”一词表明比较仅限于a附近的一个小邻域，而非整个定义域。
Over a closed interval [a, b], the largest value of f(x) is called the absolute (or global) maximum, and the smallest value is the absolute (or global) minimum. A function may possess several local extrema, but on a closed interval it has exactly one absolute maximum and one absolute minimum, though these may occur at more than one point.
在闭区间[a, b]上，f(x)的最大值称为绝对（或全局）最大值，最小值称为绝对（或全局）最小值。函数可以有多个局部极值，但在闭区间上只有一个绝对最大值和一个绝对最小值，尽管它们可能在多个点处取得。
2. Stationary Points and Critical Points | 驻点与临界点
A stationary point is a point on the curve where the derivative is exactly zero, that is, f′(x) = 0. Geometrically, the tangent line at a stationary point is horizontal. Typical examples include the vertex of a parabola, the top of a cubic “hill”, and the bottom of a cubic “valley”.
驻点是曲线上导数恰好为零的点，即f′(x) = 0。从几何上看，驻点处的切线是水平的。典型例子包括抛物线的顶点、三次函数”山峰”的顶部和”山谷”的底部。
A critical point is any point in the domain of f where either f′(x) = 0 or f′(x) does not exist. Every stationary point is a critical point, but the converse is not true. In IB examinations, you must also consider points where the derivative is undefined, such as cusps or vertical tangent points, because extrema can occur there.
临界点是定义域内使得f′(x) = 0或f′(x)不存在的点。每个驻点都是临界点，但反之不成立。在IB考试中，必须同时考虑导数不存在的点，例如尖点或垂直切点，因为极值也可能在这些点处取得。
3. The First Derivative Test | 一阶导数判定法
The first derivative test is the most reliable method for classifying extrema. Suppose x = a is a critical point. Examine the sign of f′(x) on both sides of a:
一阶导数判定法是最可靠的极值分类方法。设x = a是临界点，考察f′(x)在a两侧的符号变化：
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If f′ changes from positive to negative as x passes through a, then f has a local maximum at x = a.
若f′随x经过a由正变负，则f在x = a处取得局部极大值。
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If f′ changes from negative to positive as x passes through a, then f has a local minimum at x = a.
若f′随x经过a由负变正，则f在x = a处取得局部极小值。
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If there is no sign change, then x = a is not an extremum; it may be a horizontal point of inflection.
若符号没有变化，则x = a不是极值点，它可能是水平拐点。
| Sign change of f′(x) at x = a | Conclusion |
| + to − | Local maximum |
| − to + | Local minimum |
| No change | Not an extremum |
This test is especially useful when the second derivative is difficult to compute or when f″(a) = 0 leaves the result inconclusive.
该判定法在二阶导数难以计算或f″(a) = 0导致结论不确定时尤为有用。
4. The Second Derivative Test | 二阶导数判定法
The second derivative test provides a quick way to classify a stationary point, provided f″(x) exists and is continuous near x = a. If f′(a) = 0, then:
二阶导数判定法为驻点的分类提供了一种快捷途径，前提是f″(x)在x = a附近存在且连续。若f′(a) = 0，则：
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If f″(a) > 0, the curve is concave up, so f has a local minimum at x = a.
若f″(a) > 0，曲线凹向上，故f在x = a处取得局部极小值。
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If f″(a) < 0, the curve is concave down, so f has a local maximum at x = a.
若f″(a) < 0，曲线凹向下，故f在x = a处取得局部极大值。
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If f″(a) = 0, the test is inconclusive; you must fall back on the first derivative test.
若f″(a) = 0，则判定失效，必须改用一阶导数判定法。
f″(a) > 0 → local minimum; f″(a) < 0 → local maximum; f″(a) = 0 → inconclusive
f″(a) > 0 → 局部极小值；f″(a) < 0 → 局部极大值；f″(a) = 0 → 无法判定
Remember that the second derivative test applies only to stationary points; it cannot be used at critical points where f′(x) is undefined.
请记住，二阶导数判定法仅适用于驻点；对于导数不存在的临界点，该方法不适用。
5. A Step-by-Step Procedure | 求极值的分步流程
To locate and classify all extrema of a differentiable function, follow this systematic procedure:
为求取并判定一个可导函数的所有极值，请遵循以下系统性步骤：
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Find the derivative f′(x).
求出导数f′(x)。
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Solve the equation f′(x) = 0 to find all stationary points. Also identify any points in the domain where f′ is undefined.
解方程f′(x) = 0，找出所有驻点；同时找出定义域内使得f′不存在的点。
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Classify each critical point using the first derivative test or the second derivative test.
用一阶导数判定法或二阶导数判定法对每个临界点进行分类。
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If the question asks for the absolute extremum on a closed interval, evaluate f at every critical point and at both endpoints, then compare the values.
若题目要求闭区间上的绝对极值，则计算f在所有临界点及两个端点处的值，再进行比较。
This four-step strategy covers both classification and optimisation questions in IB Paper 1 and Paper 2.
这一四步策略可覆盖IB卷一与卷二中的分类和优化问题。
6. Local Extrema vs Global Extrema | 局部极值与全局极值
A frequent source of confusion is the difference between local and global extrema. A local extremum is defined by comparing f(a) with nearby values only. A global extremum on an interval [a, b] compares f(c) with every value in that interval. The “closed interval method” states that the absolute maximum and minimum of a continuous function on a closed interval occur either at a critical point or at an endpoint.
常见的混淆来源是局部极值与全局极值的区别。局部极值仅通过将f(a)与邻近值比较来定义；而区间[a, b]上的全局极值需要将f(c)与该区间内的所有值比较。”闭区间法”指出：闭区间上连续函数的绝对最大值与最小值，要么出现在临界点处，要么出现在端点处。
Example: Consider f(x) = 2x³ − 3x² − 12x + 5 on the interval [−2, 3]. Since f′(x) = 6x² − 6x − 12 = 6(x − 2)(x + 1), the critical points are x = −1 and x = 2. Evaluating:
示例：考虑区间[−2, 3]上的函数f(x) = 2x³ − 3x² − 12x + 5。由于f′(x) = 6x² − 6x − 12 = 6(x − 2)(x + 1)，临界点为x = −1与x = 2。计算各点函数值：
| x | −2 | −1 | 2 | 3 |
| f(x) | 1 | 12 | −15 | −4 |
Hence the absolute maximum is 12 at x = −1, and the absolute minimum is −15 at x = 2. Note that x = −1 gives only a local maximum when considered over the whole real line, but on this restricted interval it is also the global maximum.
因此绝对最大值为x = −1处的12，绝对最小值为x = 2处的−15。注意，在整个实数范围内，x = −1只是局部极大值点；但在该限定区间上，它同时也是全局最大值点。
7. Optimisation Applications | 优化应用
Optimisation is one of the most heavily tested applications of extrema in IB Mathematics. The general strategy is to express the quantity to be optimised as a function of a single variable, differentiate, set the derivative to zero, and verify that the solution indeed gives the required extremum.
优化是IB数学中极值应用的重头戏。通用策略是：将待优化的量表示为单变量函数，求导，令导数为零，并验证该解确实给出所求的极值。
Classic example: A farmer has 100 metres of fencing and wishes to enclose a rectangular field using one side of a long straight wall as a boundary. Let the two perpendicular sides each have length x; then the side parallel to the wall has length 100 − 2x, and the area is A(x) = x(100 − 2x) = 100x − 2x². Differentiating gives A′(x) = 100 − 4x. Setting A′(x) = 0 yields x = 25. Since A″(x) = −4 < 0, this is a local maximum, and the maximum area is A(25) = 25 × 50 = 1250 m².
经典例题：一位农夫有100米篱笆，想借助一堵笔直的围墙围出一块矩形场地。设两条垂直于墙的边各长x，则平行于墙的边长为100 − 2x，面积为A(x) = x(100 − 2x) = 100x − 2x²。求导得A′(x) = 100 − 4x。令A′(x) = 0得x = 25。由于A″(x) = −4 < 0，这是局部极大值，最大面积为A(25) = 25 × 50 = 1250平方米。
In any optimisation problem, remember four checks: the domain of the variable, the endpoints of the domain, the units of the answer, and the justification that the extremum is indeed a maximum or a minimum.
在任何优化题中，请记住四项检查：变量的定义域、定义域的端点、答案的单位，以及对极值确为最大值或最小值的论证。
8. Extrema and Curve Sketching | 极值与函数作图
Extrema play a key role in sketching graphs of polynomial and rational functions. The sign of f′ determines where the function is increasing or decreasing: if f′ > 0 on an interval, f is increasing there; if f′ < 0, f is decreasing. Local extrema occur exactly where this monotonic behaviour reverses.
极值在多项式函数与有理函数的作图中起着关键作用。f′的符号决定了函数的增减性：若在某区间上f′ > 0，则f在该区间递增；若f′ < 0，则f递减。局部极值恰好出现在这种单调性发生反转的地方。
For a cubic function f(x) = ax³ + bx² + cx + d with a > 0, the graph typically rises from −∞, reaches a local maximum, descends to a local minimum, then rises to +∞. Knowing the coordinates of these two turning points allows you to sketch the curve accurately without plotting many points.
对于a > 0的三次函数f(x) = ax³ + bx² + cx + d，图像通常从−∞上升，达到局部极大值，然后下降至局部极小值，再上升至+∞。知道这两个转向点的坐标，就可以不描太多点而较准确地画出曲线。
When sketching, always mark the stationary points with their exact coordinates, indicate the nature (max or min), and label the axes clearly. In IB Paper 2, a poorly labelled sketch may lose method marks even if the algebra is correct.
作图时，务必标出驻点的精确坐标，注明是极大值还是极小值，并清晰标注坐标轴。在IB卷二中考时，即使代数正确，作图标注不清也可能丢失方法分。
9. Common Pitfalls and Misconceptions | 常见误区与易错点
The following misconceptions frequently cost IB students marks in examinations. Master them to avoid unnecessary errors.
以下误区常常导致IB学生在考试中丢分。掌握它们，可以避免不必要的错误。
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f′(a) = 0 does not guarantee an extremum. For f(x) = x³, we have f′(0) = 0, yet x = 0 is a horizontal point of inflection, not a maximum or minimum, because the sign of f′ does not change.
f′(a) = 0并不保证是极值。对f(x) = x³，有f′(0) = 0，但x = 0是水平拐点而非极值点，因为f′的符号没有改变。
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f″(a) = 0 is not necessarily a point of inflection. For f(x) = x⁴, we have f″(0) = 0, but x = 0 is actually a local minimum, since f′(x) changes from negative to positive there.
f″(a) = 0未必是拐点。对f(x) = x⁴，有f″(0) = 0，但
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