Stationary Points of Multivariable Functions | 多元函数驻点的判定方法

📚 Stationary Points of Multivariable Functions | 多元函数驻点的判定方法

In multivariable calculus, finding stationary points of a function helps us locate local maxima, local minima, and saddle points. For a function of two variables, the process involves solving the first partial derivative equations and then applying a second-derivative test based on the Hessian matrix.

在多元微积分中，求出多元函数的驻点可以帮助我们定位局部极大值、局部极小值以及鞍点。对于二元函数而言，这一过程需要先求解一阶偏导数方程，再利用基于 Hessian 矩阵的二阶导数判别法进行判定。

1. Definition of Stationary Points | 驻点的定义

For a function \(f(x,y)\), a stationary point occurs where both first partial derivatives are zero simultaneously:

对于函数 \(f(x,y)\)，驻点出现在两个一阶偏导数同时为零的位置：

fₓ = ∂f/∂x = 0 且 fᵧ = ∂f/∂y = 0

Geometrically, this means the tangent plane to the surface is horizontal. The function may have a local maximum, a local minimum, or neither (a saddle point).

从几何上看，这意味着曲面在该点处的切平面是水平的。此时函数可能具有局部极大值、局部极小值，也可能两者都不是（即鞍点）。

2. Finding Critical Points | 求临界点

To find stationary points, solve the system of equations:

为求驻点，需要解方程组：

fₓ(x, y) = 0
fᵧ(x, y) = 0

Each solution \((a, b)\) is a critical point. Note that not every critical point is a stationary point of extremal type — further classification is needed.

每一个解 \((a, b)\) 都是一个临界点。注意并非每个临界点都是极值型驻点——还需要进一步分类。

3. Second-Order Partial Derivatives | 二阶偏导数

After obtaining a critical point, we compute the second-order partial derivatives:

得到临界点后，我们需要计算二阶偏导数：

fₓₓ — differentiate fₓ with respect to x again.
fᵧᵧ — differentiate fᵧ with respect to y again.
fₓᵧ — differentiate fₓ with respect to y (or fᵧ with respect to x).

fₓₓ —— 再次对 x 求偏导。
fᵧᵧ —— 再次对 y 求偏导。
fₓᵧ —— 先对 x 再对 y 求偏导（或先对 y 再对 x）。

For most standard functions, the mixed partial derivatives are equal (Clairaut’s theorem), so fₓᵧ = fᵧₓ.

对于通常的函数，混合偏导相等（Clairaut 定理），因此 fₓᵧ = fᵧₓ。

4. The Hessian Matrix | Hessian 矩阵

The Hessian matrix organizes these second-order derivatives:

Hessian 矩阵将这些二阶导数组织起来：

H = [ fₓₓ fₓᵧ ; fᵧₓ fᵧᵧ ]

The determinant of H, often denoted D, plays a central role in the classification:

H 的行列式通常记为 D，它在分类中起到核心作用：

D = fₓₓ fᵧᵧ − (fₓᵧ)²

Note: In the IB syllabus, the notation is usually fₓₓ, fᵧᵧ, fₓᵧ, and the discriminant D = fₓₓ fᵧᵧ − (fₓᵧ)².

注意：在 IB 课程大纲中，通常使用记号 fₓₓ、fᵧᵧ、fₓᵧ，以及判别式 D = fₓₓ fᵧᵧ − (fₓᵧ)²。

5. Second Derivative Test | 二阶导数判别法

At a critical point (a, b), compute D = fₓₓ fᵧᵧ − (fₓᵧ)². Then:

在临界点 (a, b) 处，计算 D = fₓₓ fᵧᵧ − (fₓᵧ)²。然后：

Condition	Conclusion
D > 0 and fₓₓ > 0	Local minimum
D > 0 and fₓₓ < 0	Local maximum
D < 0	Saddle point
D = 0	Test is inconclusive

In the case D = 0, higher-order derivatives or other methods must be used to determine the nature of the point.

当 D = 0 时，判别法失效，需要借助高阶导数或其他方法判断该点的性质。

6. Worked Example 1 — Local Minimum | 实例 1 —— 局部极小值

Consider \(f(x,y) = x² + y² − 2x − 4y + 5\).

考察 \(f(x,y) = x² + y² − 2x − 4y + 5\)。

First partial derivatives:

先求一阶偏导数：

fₓ = 2x − 2 = 0 → x = 1
fᵧ = 2y − 4 = 0 → y = 2

Thus, the only critical point is (1, 2).

因此唯一临界点为 (1, 2)。

Second derivatives:

二阶导数：

fₓₓ = 2, fᵧᵧ = 2, fₓᵧ = 0

Therefore D = (2)(2) − 0² = 4 > 0, and fₓₓ = 2 > 0. This confirms a local minimum.

因此 D = (2)(2) − 0² = 4 > 0，且 fₓₓ = 2 > 0。这确认了该点为局部极小值。

7. Worked Example 2 — Saddle Point | 实例 2 —— 鞍点

Consider \(f(x,y) = x² − y²\).

考察 \(f(x,y) = x² − y²\)。

Solving fₓ = 2x = 0 and fᵧ = −2y = 0 gives (0, 0).

由 fₓ = 2x = 0 和 fᵧ = −2y = 0 解得唯一临界点 (0, 0)。

Second derivatives:

二阶导数：

fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0

Thus D = (2)(−2) − 0² = −4 < 0, so (0, 0) is a saddle point.

于是 D = (2)(−2) − 0² = −4 < 0，因此 (0, 0) 是鞍点。

This function increases along the x-direction and decreases along the y-direction, resembling a horse’s saddle.

该函数沿 x 方向上升、沿 y 方向下降，形状如同马鞍。

8. Worked Example 3 — Inconclusive D = 0 | 实例 3 —— D = 0 失效情形

Consider \(f(x,y) = x⁴ + y⁴\).

考察 \(f(x,y) = x⁴ + y⁴\)。

At (0, 0), both first derivatives vanish. The second derivatives at (0, 0) are all 0, so D = 0.

在 (0, 0) 处，两个一阶偏导数均为零。而二阶偏导数在 (0, 0) 处也都为零，因此 D = 0。

Using the second derivative test is inconclusive. However, since x⁴ ≥ 0 and y⁴ ≥ 0, the function has an absolute minimum at (0, 0). This shows that D = 0 does not automatically mean “not a local extremum”.

二阶导数判别法失效。但由于 x⁴ ≥ 0 且 y⁴ ≥ 0，函数在 (0, 0) 处具有绝对极小值。这说明了 D = 0 并不意味着“一定不是极值”。

9. Relationship with the Hessian Matrix Eigenvalues | 与 Hessian 矩阵特征值的关系

An alternative criterion uses the eigenvalues of the Hessian matrix:

另一种判据利用 Hessian 矩阵的特征值：

If both eigenvalues are positive → local minimum.
If both eigenvalues are negative → local maximum.
If eigenvalues have opposite signs → saddle point.
If any eigenvalue is zero → inconclusive.

若两个特征值均为正 → 局部极小值。
若两个特征值均为负 → 局部极大值。
若特征值异号 → 鞍点。
若存在特征值为零 → 判别失效。

This perspective is especially useful in higher dimensions and in optimization theory.

这一视角在高维和优化理论中尤其有用。

10. Common Mistakes and Tips | 常见错误与建议

Here are some frequent errors to avoid in exams:

以下是考试中需要避免的常见错误：

Forgetting to check both fₓ = 0 and fᵧ = 0 before applying the second derivative test.
Using \(f_{xy}\) instead of \((f_{xy})²\) in the discriminant formula.
Confusing the sign conditions for local maxima and minima.
Assuming D = 0 means a saddle point — it is actually inconclusive.

在应用二阶导数判别法之前忘记同时检验 fₓ = 0 和 fᵧ = 0。
在判别式公式中把 fₓᵧ 误用为 fₓᵧ 的平方。
混淆局部极大值与局部极小值关于 fₓₓ 符号的条件。
误以为 D = 0 就是鞍点——实际上判别法失效。

Always write down all partial derivatives clearly and state which conclusion each condition gives.

务必清晰地写出各偏导数，并说明每个条件对应的结论。

11. Summary | 总结

The classification of stationary points for a two-variable function follows a clear procedure:

二元函数驻点的分类遵循一个清晰的流程：

Solve fₓ = 0 and fᵧ = 0 to find critical points.
Compute fₓₓ, fᵧᵧ, and fₓᵧ at each critical point.
Calculate D = fₓₓ fᵧᵧ − (fₓᵧ)².
Apply the sign table to classify.

解方程 fₓ = 0 和 fᵧ = 0 求临界点。
在每个临界点处计算 fₓₓ、fᵧᵧ 和 fₓᵧ。
计算 D = fₓₓ fᵧᵧ − (fₓᵧ)²。
根据符号表进行分类。

Mastering this method is essential for solving optimization problems in IB Mathematics HL and beyond.

掌握这个方法对解决 IB 数学 HL 及更高阶的优化问题至关重要。

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Stationary Points of Multivariable Functions | 多元函数驻点的判定方法