Solving and Identifying Extreme Points | 极值点的求解与判断

📚 Solving and Identifying Extreme Points | 极值点的求解与判断

In IB Mathematics, finding and classifying extreme points (also called turning points or local extrema) is a core skill in both Analysis and Approaches and Applications and Interpretation. Extreme points tell us where a function reaches a local maximum or minimum, and they are essential for curve sketching, optimisation problems, and interpreting real-world models.

在IB数学中，求解与判断极值点（也称为转折点或局部最值）是分析与方法（AA）以及应用与解释（AI）两门课程的核心技能。极值点帮助我们确定函数在局部取得最大值或最小值的位置，也是绘制函数图像、解决最优化问题以及解读现实模型的基础。

1. What Is an Extreme Point? | 什么是极值点？

An extreme point of a function is a point where the function value is either a local maximum or a local minimum. More precisely, f(c) is a local maximum if f(c) ≥ f(x) for all x sufficiently close to c, and a local minimum if f(c) ≤ f(x) for all x sufficiently close to c.

函数的极值点是指函数在该点的值为局部最大值或局部最小值的位置。更精确地说，如果对于所有足够接近c的x，有f(c) ≥ f(x)，则f(c)是局部最大值；如果f(c) ≤ f(x)，则f(c)是局部最小值。

Local maximum / 局部最大值
Local minimum / 局部最小值
Global maximum / minimum over a given interval / 给定区间上的全局最大值与最小值

2. The Necessary Condition: Critical Points | 必要条件：临界点

If a function f is differentiable at x = c and has a local extremum at c, then f'(c) = 0. This is known as Fermat’s theorem. However, the converse is not true: f'(c) = 0 does not guarantee that c is an extreme point.

如果函数f在x = c处可导，并且在c处取得局部极值，那么必有f'(c) = 0。这就是费马定理。但反过来并不成立：f'(c) = 0并不能保证c一定是极值点。

Critical points are points where f'(x) = 0 or where f'(x) does not exist. All candidate extreme points must be found among these critical points.

临界点是使得f'(x) = 0或f'(x)不存在的点。所有可能的极值点都必须在这些临界点中寻找。

f'(c) = 0 或 f'(c) 不存在

3. First Derivative Test | 一阶导数判别法

The first derivative test examines the sign of f'(x) on the left and right of a critical point. If f'(x) changes from positive to negative, the critical point is a local maximum. If it changes from negative to positive, it is a local minimum. If the sign does not change, the point is neither a maximum nor a minimum.

一阶导数判别法考察临界点左右两侧f'(x)的符号。如果f'(x)由正变负，则该临界点为局部最大值；如果由负变正，则为局部最小值；如果符号不变，则该点既不是最大值也不是最小值。

f'(x): + → − : local maximum / 局部最大值
f'(x): − → + : local minimum / 局部最小值
f'(x): + → + or − → − : no extremum / 没有极值

4. Second Derivative Test | 二阶导数判别法

For a critical point c where f'(c) = 0 and f”(c) exists, the second derivative test is often quicker. If f”(c) > 0, the function is concave up and c is a local minimum. If f”(c) < 0, the function is concave down and c is a local maximum. If f''(c) = 0, the test is inconclusive.

对于满足f'(c) = 0且f”(c)存在的临界点c，二阶导数判别法通常更快捷。如果f”(c) > 0，函数向上凸（凹向上），c为局部最小值；如果f”(c) < 0，函数向下凸，c为局部最大值；如果f''(c) = 0，则无法判断。

f”(c) > 0 → 极小值；f”(c) < 0 → 极大值；f''(c) = 0 → 无法判断

When f”(c) = 0, you must use the first derivative test or investigate higher derivatives.

当f”(c) = 0时，你必须改用一阶导数判别法或进一步考察更高阶导数。

5. Solving Steps for Extreme Points | 求解极值点的步骤

The standard procedure for finding and classifying extreme points can be summarised in five steps.

求解和判断极值点的标准流程可以概括为五个步骤。

Find f'(x) by differentiation. / 通过求导得到f'(x)。
Solve f'(x) = 0 and also check where f'(x) is undefined to locate critical points. / 解方程f'(x) = 0，同时检查f'(x)不存在的点，以找到所有临界点。
Determine the sign of f'(x) on intervals between critical points, or evaluate f”(x) at each critical point. / 判断临界点之间区间上f'(x)的符号，或在每个临界点计算f”(x)。
Classify each critical point as a local maximum, local minimum, or neither. / 将每个临界点分类为局部最大值、局部最小值或不是极值。
For global extrema on a closed interval, compare values at critical points and endpoints. / 对于闭区间上的全局最值，比较临界点和端点的函数值。

6. Example: Quadratic Function | 例题：二次函数

Consider f(x) = x² − 4x + 3. Its derivative is f'(x) = 2x − 4. Setting f'(x) = 0 gives x = 2. The second derivative is f”(x) = 2 > 0, so x = 2 is a local minimum.

考虑f(x) = x² − 4x + 3。其导数为f'(x) = 2x − 4。令f'(x) = 0，得x = 2。二阶导数f”(x) = 2 > 0，因此x = 2是局部最小值。

f'(x) = 2x − 4 = 0 → x = 2, f(2) = −1

Since f”(2) = 2 > 0, the point (2, −1) is a local minimum. Because the leading coefficient is positive, this is also the global minimum.

由于f”(2) = 2 > 0，点(2, −1)是局部最小值。又因为二次项系数为正，该点同时也是全局最小值。

7. Example: Cubic Function | 例题：三次函数

Let f(x) = x³ − 3x² + 2. Then f'(x) = 3x² − 6x = 3x(x − 2). Critical points are x = 0 and x = 2. Using the first derivative test:

设f(x) = x³ − 3x² + 2。则f'(x) = 3x² − 6x = 3x(x − 2)。临界点为x = 0和x = 2。使用一阶导数判别法：

Interval	(−∞, 0)	(0, 2)	(2, ∞)
Sign of f'(x)	+	−	+

The sign changes from + to − at x = 0, giving a local maximum f(0) = 2. At x = 2, the sign changes from − to +, giving a local minimum f(2) = −2.

在x = 0处，符号由正变负，因此得到局部最大值f(0) = 2。在x = 2处，符号由负变正，因此得到局部最小值f(2) = −2。

f(0) = 2（极大值）, f(2) = −2（极小值）

8. Example: Exponential Function | 例题：指数函数

Consider f(x) = x·e⁻ˣ. Using the product rule, f'(x) = e⁻ˣ − x·e⁻ˣ = e⁻ˣ(1 − x). Since e⁻ˣ > 0 for all x, f'(x) = 0 gives x = 1.

考虑f(x) = x·e⁻ˣ。用乘积法则求导，f'(x) = e⁻ˣ − x·e⁻ˣ = e⁻ˣ(1 − x)。由于e⁻ˣ > 0对所有x成立，令f'(x) = 0得x = 1。

Now f”(x) = −e⁻ˣ − (e⁻ˣ − x·e⁻ˣ) = e⁻ˣ(x − 2). At x = 1, f”(1) = −e⁻¹ < 0, so x = 1 is a local maximum.

再求f”(x) = −e⁻ˣ − (e⁻ˣ − x·e⁻ˣ) = e⁻ˣ(x − 2)。在x = 1处，f”(1) = −e⁻¹ < 0，因此x = 1是局部最大值。

f(1) = 1·e⁻¹ = e⁻¹ (局部最大值)

9. Global Extrema on a Closed Interval | 闭区间上的全局最值

When finding global (absolute) extrema of a continuous function on a closed interval [a, b], you must evaluate f at all critical points inside the interval and also at the endpoints a and b. The largest value is the global maximum; the smallest is the global minimum.

当在闭区间[a, b]上寻找连续函数的全局（绝对）最值时，你必须计算区间内所有临界点以及端点a和b处的函数值。最大值为全局最大值，最小值为全局最小值。

Example: f(x) = x³ − 3x² + 2 on [−1, 3]. Critical points are x = 0, 2. Endpoints: f(−1) = −2, f(3) = 2. Critical values: f(0) = 2, f(2) = −2. Thus global maximum is 2 and global minimum is −2.

例如：在区间[−1, 3]上考虑f(x) = x³ − 3x² + 2。临界点为x = 0, 2。端点值：f(−1) = −2，f(3) = 2。临界值：f(0) = 2，f(2) = −2。因此全局最大值为2，全局最小值为−2。

10. Points Where the Derivative Does Not Exist | 导数不存在的点

In IB problems, extreme points can also occur where f'(x) is undefined, such as at a corner or a vertical tangent. For example, f(x) = |x| has a local minimum at x = 0 even though f'(0) does not exist. You must always check these points as candidates.

在IB题目中，极值点也可能出现在f'(x)不存在的点，例如尖点或竖直切线处。例如，f(x) = |x|在x = 0处有局部最小值，尽管f'(0)不存在。你必须始终将这些点作为候选点进行检查。

If f'(c) is undefined but f(c) is defined, test the sign change of f'(x) around c.

如果f'(c)不存在但f(c)有定义，需要检验f'(x)在c附近的符号变化。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Students often lose marks by forgetting to classify critical points or by ignoring endpoints. Below are common pitfalls and how to avoid them.

学生常常因为忘记判断临界点的类型或忽略端点而失分。以下是一些常见陷阱及应对方法。

Mistake: Assuming f'(c) = 0 always implies an extremum. Counterexample: f(x) = x³ at x = 0 has f'(0) = 0 but no extremum. / 错误：认为f'(c) = 0一定意味着极值。反例：f(x) = x³在x = 0处f'(0) = 0，但没有极值。
Mistake: Using the second derivative test when f”(c) = 0 and concluding no extremum. The test is inconclusive; use the first derivative test. / 错误：当f”(c) = 0时使用二阶导数判别法并断定没有极值。此时该判别法无效，应使用一阶导数判别法。
Mistake: Forgetting to check endpoints when finding global extrema on a closed interval. / 错误：在闭区间上求全局最值时忘记检查端点。
Tip: Always show the sign table for the first derivative test to communicate your reasoning clearly. / 提示：始终写出一阶导数的符号表，以清晰展示推理过程。

12. Summary | 总结

To solve and identify extreme points, follow the logical chain: differentiate, locate critical points, classify using the first or second derivative test, and, if required, compare endpoint values for global extrema. Mastery of these steps will help you handle polynomial, trigonometric, exponential, and rational functions confidently in IB exams.

求解与判断极值点需要遵循一个逻辑链条：求导、找临界点、用一阶或二阶导数判别法分类，如果需要的话，再比较端点值以确定全局最值。熟练掌握这些步骤，你将能在IB考试中从容处理多项式、三角函数、指数函数和有理函数的相关问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving and Identifying Extreme Points | 极值点的求解与判断