📚 3 Exceptional Cases of Maxima and Minima | 极大值与极小值的三种例外情况
nd minima of a function is a fundamental skill. We usually start by finding stationary points where f' ative test. However, this textbook recipe can miss critical cases. A function may achieve its highest or lowest value at points where the derivative does not exist, at stationary points that are not extrema, or at boundary endpoints of a closed interval. Missing any of these three exceptional cases can lead to incorrect conclusions, especially in optimisation problems. This article systematically analyses these three exceptional cases, shows how to handle them with the first derivative test, and reinforces the concepts through worked examples relevant to the IB Analysis & Approaches and Applications & Interpretation syllabuses.
在IB数学中，寻找函数的极大值与极小值是一项基本技能。我们通常先求导数为零的驻点 f'(x)=0，再运用二阶导数判别法。然而，这一标准的操作流程可能遗漏关键情形。函数可能在导数不存在的点、驻点但非极值点、或闭区间的边界端点处取得最大值或最小值。忽略这三种例外情况中的任何一种，都可能导致错误的结论，尤其在优化问题中更是如此。本文系统分析这三种例外情形，展示如何用一阶导数判别法加以处理，并结合IB分析数学与应用数学课程中的典型例题来巩固概念。
1. Standard Approach to Extrema | 寻找极值的标准方法
n f on an open interval, a local maximum or minimum occurs only at critical points where f'(x)=0. The second derivative test then provides a convenient classification: if f'(c)=0 and f''(c)>0, f has a local minimum at c; if f'(c)=0 and f'' blems and carefully designed IB exam questions often introduce wrinkles that break these neat rules.
对于开区间上的可微函数 f，局部极大值或极小值只可能出现在 f'(x)=0 的临界点处。二阶导数判别法则给出便捷的分类：若 f'(c)=0 且 f''(c)>0，则 f 在 c 处取得局部极小值；若 f'(c)=0 且 f''(c)<0，则 f 在 c 处取得局部极大值。这一流畅的过程对多项式及许多性质良好的函数非常有效。然而，实际问题以及精心设计的IB考题经常会引入打破这些规整法则的细节。
A more complete definition insists that local extrema can occur at any critical point – any x-value in the domain of f where f'(x)=0 or f' ined, yet global extrema often live there. This expanded perspective naturally leads to the three exceptional cases that students must master.
更完整的定义强调，局部极值可能出现在任何临界点——即函数定义域内 f'(x)=0 或 f'(x) 不存在的任意 x 值。此外，在闭区间 [a, b] 上，我们必须始终计算 f(a) 与 f(b)，因为端点处导数无定义，而全局极值往往居于此处。这一扩展的视角自然地引出了学生必须掌握的三种例外情况。
2. Exceptional Case 1: Points Where the Derivative Does Not Exist | 例外情况一：导数不存在的点
tangent, and yet attain a local maximum or minimum at that very point. For example, f(x)=|x| has a corner at x=0. The derivative f'(x) is -1 for x<0 and +1 for x>0, so f' global minimum. Ignoring x=0 as a critical point would mean missing the extremum entirely.
最容易被忽视的临界点是那些导数不存在的点。函数可能存在尖角、尖点或垂直切线，却恰好在该点取得局部极大值或极小值。例如，f(x)=|x| 在 x=0 处有一个角点。导数 f'(x) 在 x<0 时为 -1，在 x>0 时为 +1，因此 f'(0) 不存在。但 f(0)=0 却是全局极小值。若忽略 x=0 作为临界点，就将完全错过这一极值。
Another instructive example is f(x)=x2/3. The derivative is f'(x)=(2/3)x-1/3, which is undefined at x=0 (vertical tangent). A sign analysis of f' h non-differentiable points, especially when piecewise or absolute-value functions appear.
另一个有启发性的例子是 f(x)=x2/3。其导数 f'(x)=(2/3)x-1/3，在 x=0 处无定义（垂直切线）。对 f'(x) 的符号分析表明，当 x<0 时导数为负，当 x>0 时导数为正，从而确认在 x=0 处取得局部极小值。IB考试的题目常常将极值隐藏于这类不可导点，尤其是在分段函数或绝对值函数出现时。
3. Exceptional Case 2: Stationary Points That Are Not Extrema | 例外情况二：非极值点的驻点
int of inflection with a horizontal tangent. The classic example is f(x)=x³ at x=0. Here f'(0)=0 and f'' on, not an extremum. If a student blindly applies the second derivative test and obtains f''(c)=0, the test is inconclusive, and the point might or might not be an extremum.
并非所有导数为零的点都是极大值或极小值。驻点可以是水平拐点。经典的例子是 f(x)=x³ 在 x=0 处的表现。此时 f'(0)=0 且 f''(0)=0，但函数在整个实数域上单调递增；x=0 是一个水平拐点，而不是极值点。如果学生盲目套用二阶导数判别法，得到 f''(c)=0，此时判别法失效，该点可能是也可能不是极值点。
To resolve this, we must use the first derivative test. For f(x)=x³, f'(x)=3x² is positive on both sides of zero, so no sign change occurs – hence no extremum. Contrast this with f(x)=x⁴, where f'(0)=0, f'' between x³ and x⁴ at the origin, which is a perfect illustration of this exceptional case.
要解决这个问题，我们必须使用一阶导数判别法。对于 f(x)=x³，f'(x)=3x² 在零的两侧均为正，没有发生符号变化——因此无极值。与之对照，f(x)=x⁴ 在 x=0 处也有 f'(0)=0 且 f''(0)=0，但一阶导数由负变正，从而得到局部极小值。仅凭二阶导数判别法无法在原点区分 x³ 与 x⁴，这正是这一例外情况的绝佳例证。
4. Exceptional Case 3: Endpoint Extrema | 例外情况三：端点极值
ng interior critical points. For instance, consider f(x)=x²-4x on [0, 5]. The derivative f' s that the absolute minimum is -4 (interior) and the absolute maximum is 5 (endpoint x=5).
当函数定义在闭区间 [a, b] 上时，由极值定理保证的绝对最大值与绝对最小值可能出现在端点，而端点处的导数概念并不以同样方式适用。学生们经常在找到内部临界点后忘记计算 f(a) 和 f(b)。例如，考虑 f(x)=x²-4x 在区间 [0, 5] 上。导数 f'(x)=2x-4 给出驻点 x=2，f(2)=-4。但计算 f(0)=0 与 f(5)=5 后便知，绝对极小值为 -4（内部点），而绝对最大值为 5（端点 x=5）。
Endpoint extrema are particularly common in optimisation problems where a physical constraint naturally restricts the domain, such as a volume problem with positive lengths. The derivative may never be zero inside the feasible interval; the maximum could occur exactly at one of the bounds. Treating endpoints as critical candidates is therefore non-negotiable for any complete extremum analysis in IB exams.
端点极值在优化问题中尤其常见，例如长度必须为正的体积问题，其物理约束自然地限制了定义域。导数在可行区间内可能恒不为零；最大值可能恰在某个边界处取得。因此，将端点视为临界候选点，对于IB考试中任何完整的极值分析而言，都是不容妥协的步骤。
5. The Second Derivative Test and Its Limitations | 二阶导数判别法及其局限性
The second derivative test is a shortcut, not a universal tool. It requires f'' to exist and be non-zero at the stationary point. If f''(c)=0, the test gives no information – be a local minimum (x⁴), a local maximum (-x⁴), or an inflection (x³). Additionally, if f''(c) does not exist, the test cannot be applied at all. Relying solely on f'' can produce false negatives or misclassifications in these exceptional scenarios.
二阶导数判别法是一条捷径，而非万能工具。它要求 f'' 在驻点处存在且非零。若 f''(c)=0，该判别法无法提供任何信息——该点可能是局部极小值 (x⁴)、局部极大值 (-x⁴) 或拐点 (x³)。此外，若 f''(c) 根本不存在，则此法完全不能应用。仅依赖 f'' 在这些例外情形下可能导致假阴性或错误分类。
tationary points where the second derivative vanishes, explicitly testing whether candidates know to switch to the first derivative test. A sign table for f' he f' around the stationary point always yields a definitive answer, regardless of the second derivative' ld be viewed as a convenient option when it works, but never as the only available method.
IB考官常常设计如 f(x)=x⁴−4x³+6x² 这样的函数，其驻点处二阶导数为零，明确考查考生是否知道转而使用一阶导数判别法。围绕驻点制作 f' 的符号表，无论二阶导数的值如何，总能得到确定的答案。因此，二阶导数判别法应被视为适用时的一个便捷选项，但绝不能当作唯一可用的方法。
6. Using the First Derivative Test to Resolve Ambiguity | 使用一阶导数判别法消除歧义
The first derivative test examines the sign change of f'(x) across a critical point c. If f' changes from positive to negative at c, f has a local maximum; if from negative to positive, a local minimum; if no sign change, c is not an extremum. This test works for all types of critical points – stationary, non-differentiable, and even endpoints when considering one-sided behaviour – making it the most robust tool in an IB student's toolkit.
一阶导数判别法检验 f'(x) 在临界点 c 两侧的符号变化。若 f' 在 c 处由正变负，则 f 有局部极大值；若由负变正，则为局部极小值；若无符号变化，则 c 不是极值点。这一判别法适用于所有类型的临界点——驻点、不可导点，甚至在考虑单侧行为时也适用于端点——使其成为IB学生工具包中最可靠的工具。
a, c₁, c₂, ..., b in order; choose test points in each subinterval; evaluate the sign of f' lustrates the procedure for a function with a corner at x=0 and a stationary point at x=2.
构建符号图表十分直接：按顺序列出所有临界值 a, c₁, c₂, ..., b；在每个子区间内选取测试点；计算这些测试点处 f' 的符号。下表展示了对一个在 x=0 有角点、在 x=2 有驻点的函数执行该流程的情形。
| Interval | (-∞, 0) | (0, 2) | (2, ∞) |
| Sign of f'(x) | + | − | + |
| Behaviour of f | Increasing | Decreasing | Increasing |
From this table, x=0 is a local maximum (f' changes from + to −) and x=2 is a local minimum (f' s both the non-differentiable point and the stationary point, unifying the analysis of exceptional cases.
根据该表，x=0 为局部极大值（f' 由 + 变 −），x=2 为局部极小值（f' 由 − 变 +）。注意这一结论完全无需检查二阶导数。一阶导数判别法顺利地处理了不可导点和驻点，将例外情况的分析统一了起来。
7. Global vs. Local Extrema and Unbounded Functions | 全局极值、局部极值与无界函数
ust differentiate between local extrema found by derivatives and global extrema determined by comparing critical values and examining domain boundaries, including limits at infinity or near asymptotes.
另一种例外情形是，函数在开区间上因趋于无穷大或存在垂直渐近线而没有全局最大值或最小值。例如，f(x)=1/x 在 (0, ∞) 上没有绝对最大值或最小值；当 x→0⁺ 时 f(x)→∞，当 x→∞ 时 f(x)→0 但永远达不到 0。学生必须区分通过导数找到的局部极值与通过比较临界值、检查定义域边界（包括无穷远处或渐近线附近的极限）而确定的全局极值。
In IB optimisation, a common pattern is a function modelling volume or cost on a realistic domain such as [0, 10]. All interior critical points and endpoints must be tested numerically. If the largest value occurs at an interior critical point but the domain changes by a small constraint, the extremum could shift to a boundary – an exceptional shift that tests deep understanding. Always stating the final answer with units and checking its physical plausibility reflects mature problem-solving.
在IB优化题中，一种常见模式是在实际定义域（如 [0, 10]）上建立体积或成本模型。所有内部临界点与端点都必须进行数值检验。若最大值出现在内部临界点，但定义域因微小约束而改变，极值可能移至边界——这种例外性的转移考验着深层次的理解。始终带单位给出最终答案，并检验其物理上的合理性，反映出成熟的解题素养。
8. Worked Example Combining All Three Exceptional Cases | 综合三种例外情况的例题
Consider g(x)=|x²-4| on the interval [-3, 3]. Write g as a piecewise function: g(x)=x²-4 for |x|≥2, and g(x)=4-x² for |x|<2. Then g'(x)=2x for |x|>2, and g'(x)=-2x for |x| nd the endpoints -3, 3. At x=-2 and x=2 the derivative does not exist (corners); at x=0, g'(0)=0; and both endpoints must be evaluated.
考虑区间 [-3, 3] 上的函数 g(x)=|x²-4|。将其写成分段形式：当 |x|≥2 时，g(x)=x²-4；当 |x|<2 时，g(x)=4-x²。于是，当 |x|>2 时 g'(x)=2x；当 |x|<2 时 g'(x)=-2x。临界点包括 x=-2, 0, 2（导数定义改变或为零之处）以及端点 -3, 3。在 x=-2 和 x=2 处导数不存在（角点）；在 x=0 处 g'(0)=0；两端点也必须评估。
| x | g(x) |
| -3 | 5 |
| -2 | 0 |
| 0 | 4 |
| 2 | 0 |
| 3 | 5 |
imum of 4, but not the global maximum. This single problem illustrates all three exceptional cases: endpoints give the global maximum, non-differentiable points give the global minimum, and a stationary point exists that is not a global extremum.
绝对最大值为 5（在两个端点处取得），绝对最小值为 0（在不可导点 x=-2 和 x=2 处取得）。驻点 x=0 给出局部极大值 4，但并非全局最大值。这一道题便展示了全部三种例外情况：端点给出了全局最大值，不可导点给出了全局最小值，而一个驻点存在却不是全局极值。
9. Common Pitfalls and How to Avoid Them | 常见陷阱及其避免方法
A frequent mistake is to differentiate, set f'(x)=0, solve, and immediately classify each solution as a maximum or minimum using f'' This skips non-differentiable points, overlooks endpoints, and may mislabel points where f''=0. To avoid this, always begin by asking: 'Is the domain a closed interval? Are there any points where f is not differentiable?' all candidate x-values before analysing. A disciplined checklist saves marks in IB exams.
一个常见错误是：求导、设 f'(x)=0、解方程，然后立即用 f'' 将每个解归类为极大值或极小值。这跳过了不可导点，忽略了端点，并可能错误标记 f''=0 的点。为避免这一问题，始终从问以下问题开始：“定义域是闭区间吗？是否存在
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