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多变量微积分

Course: 多变量微积分 > 单位 3

课程 3: 优化多元函数 (文章)
数学
多变量微积分
多变量导数的应用
优化多元函数 (文章)
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二阶偏导数测试

Google课堂
了解如何测试具有两个输入自变量的函数是否具有局部最大值或最小值。

背景知识

非绝对必要，但在一个章节中使用：
另外, 如果你对单变量微积分的二阶导数检验法有点生疏, 你可能想在这里快速回顾一下, 因为它是二阶偏导数检验法的一个很好的比较。

二阶偏导数检验法的陈述

如果你正在求两个变量函数f, left parenthesis, x, comma, y, right parenthesis的 局部最大值/最小值 , 第一步是求输入点left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, 其梯度是 start bold text, 0, end bold text向量。
del, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, equals, start bold text, 0, end bold text
这些基本上是f图上切线平面平坦的点。
二阶偏导数检验法 告诉我们如何检验此稳定点是局部最大值、局部最小值还是鞍点。 具体来说, 你首先要计算此数量:
H, equals, start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612, minus, start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0d923f, squared
然后, 二阶偏导数检验法如下所示:
  • 如果 H, is less than, 0, 则left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis是个鞍点。
  • 如果H, is greater than, 0, 则 left parenthesis, x, start subscript, 0, end subscript, 、, y, start subscript, 0, end subscript, right parenthesis 是最高或最低点, 你再问一个问题:
    • 如果start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, is less than, 0, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis 则有局部最大值.
    • 如果start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, is greater than, 0, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis 则有局部最小值.
    (你可以用 start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612 而不是 start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, 因为并没有区别)
  • 如果 H, equals, 0, 我们没有足够的信息去知道.

直觉

fxx(x0,y0)⏞凹性在x-方向fyy(x0,y0)⏞凹性在y-方向⏟正数仅当x与 y方向与凹性方向一致−fxy(x0,y0)2⏟函数f 有多少类似g(x,y)=xy\underbrace{\overbrace{\blueD{f_{xx}(x_0,y_0)}}^{\begin{array}{c}\scriptsize\text{凹性}\\\scriptsize\text{在x-方向}\end{array}}\overbrace{\redD{f_{yy}(x_0,y_0)}}^{\begin{array}{c}\scriptsize\text{凹性}\\\scriptsize\text{在y-方向}\end{array}}}_{\begin{array}{c}\scriptsize\text{正数仅当x与 y}\\\scriptsize\text{方向与凹性方向一致}\end{array}}-\underbrace{\greenD{f_{xy}(x_0,y_0)^2}}_{\begin{array}{c}\scriptsize\text{函数}f\text{ 有多少}\\\scriptsize\text{类似}g(x,y)=xy\end{array}}
首先关注这项:
start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612
你可以认为它巧妙地编码了f图形的凹性，在x和y方向上是否相同。
例如, 查看函数
f, left parenthesis, x, comma, y, right parenthesis, equals, x, squared, minus, y, squared
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此函数的鞍点为left parenthesis, x, comma, y, right parenthesis, equals, left parenthesis, 0, comma, 0, right parenthesis。 关于x的二阶偏导数是一个正常数：
fxx(x,y)=∂∂x∂∂x(x2−y2)=∂∂x2x=2>0\begin{aligned} \blueE{f_{xx}(x, y)} &= \dfrac{\partial}{\partial x}\dfrac{\partial}{\partial x} (x^2 - y^2) \\ \\ &= \dfrac{\partial}{\partial x} 2x \\ \\ &= 2 >0 \end{aligned}
尤其是,start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0c7f99, equals, 2, is greater than, 0, 这个正数意味着沿着x-方向，f, left parenthesis, x, comma, y, right parenthesis向上凹。另一方面, 对于y的第二个偏导数是个负常数：
fyy(x,y)=∂∂y∂∂y(x2−y2)=∂∂y−2y=−2<0\begin{aligned} \redE{f_{yy}(x, y)} &= \dfrac{\partial}{\partial y}\dfrac{\partial}{\partial y} (x^2 - y^2) \\\\ &= \dfrac{\partial}{\partial y} -2y \\\\ &= -2 < 0 \end{aligned}
这意味我们沿着y-方向函数是下凹。这种不匹配意味着我们有一个鞍点, 它被编码在两个二阶偏导数的乘积里面:
start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #bc2612, equals, left parenthesis, 2, right parenthesis, left parenthesis, minus, 2, right parenthesis, equals, minus, 4, is less than, 0
由于start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0d923f, squared只能是正的，因此减去它只会使整个表达式变得更负。
start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612, minus, start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0d923f, squared
另一方面, 当start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99 和 start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, y, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612符号同时是正的，或两者同时是负的, 那么沿着x和y方向对f的凹性判断一致. 在这两种情况下, 这个乘积start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612将会是正的.
但这还不够

start color #0d923f, f, start subscript, x, y, end subscript, squared, end color #0d923f 项

思考这个函数
f(x,y)=x2+y2+pxy\begin{aligned} f(x, y) = x^2 + y^2 + \greenE{p}xy \end{aligned}
start color #0d923f, p, end color #0d923f 是个常数.
概念检查 : 使用f的定义, 计算其二阶偏导数:
start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, comma, y, right parenthesis, end color #0c7f99, equals
start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, comma, y, right parenthesis, end color #bc2612, equals
start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, x, comma, y, right parenthesis, end color #0d923f, equals

因为 start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0c7f99 和 start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #bc2612的二阶偏导数都是正的, 图像在 x 方向或y 方向都会是向上凹 (不管start color #0d923f, p, end color #0d923f 是什么).
但是, 请观看下面的视频, 在视频中, 我们展示了此图形是如何变化的, 因为我们让常量 start color #0d923f, p, end color #0d923f从1变化3,然后又回到1:
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这是怎么回事？ 即使在x和y方向上都向上凹, 图形怎么可能有鞍点呢？简短的回答是, 其他方向也很重要, 在这种情况下, 它们被 start color #0d923f, p, end color #0d923f, x, y这一项所捕捉到。
例如, 如果我们将此x, y项隔离开来, 并查看 g, left parenthesis, x, comma, y, right parenthesis, equals, x, y的图形, 下面是它的外观:
g, left parenthesis, x, comma, y, right parenthesis, equals, x, y的图像.
g(x, y) = xy的图像，和图像x² - y²非常相似, 只是旋转了45°并且扩大了一点。
在left parenthesis, 0, comma, 0, right parenthesis有个鞍点。这不是因为x和y方向凹性不一致, 而是因为凹性在对角方向 [11]\left[\begin{array}{c} 1 \\ 1 \end{array} \right]是正的，但在[−11]\left[\begin{array}{c} -1 \\ 1 \end{array} \right]的方向是负的。
让我们看看二阶导数测试告诉我们的函数f, left parenthesis, x, comma, y, right parenthesis, equals, x, squared, plus, y, squared, plus, start color #0d923f, p, end color #0d923f, x, y 。使用上面要求你计算的二阶导数的值，这是我们得到的：
start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #bc2612, minus, start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0d923f, squared, equals, left parenthesis, start color #0c7f99, 2, end color #0c7f99, right parenthesis, left parenthesis, start color #bc2612, 2, end color #bc2612, right parenthesis, minus, start color #0d923f, p, end color #0d923f, squared
当 p, is greater than, 2时, 这是负面的, 因此 f有一个鞍点。 当 p, is less than, 2 时, 它是积极的, 因此 f 具有本地最低值。
你可以将数量start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0d923f视为函数f像图形g, left parenthesis, x, comma, y, right parenthesis, equals, x, y在点left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis附近的测量
考虑到有多少方向必须彼此一致, 它实际上是相当令人惊讶的, 我们只需要考虑三个值, start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #bc2612 和start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, 0, comma, 0, right parenthesis, end color #0d923f.
下篇文章 给了更多关于二阶偏导数检验法的详细解释.

总结

  • 一旦找到一个点, 其中一个多变量函数的梯度是零向量, 这意味着图形的切线平面在这个点是平的, 二阶偏导数测试是一种方法, 以判断该点是否为局部最大值, 局部最小值, 或鞍点。
  • 二阶偏导数检验法的关键项是：
H, equals, start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, start color #bc2612, f, start subscript, y, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612, minus, start color #0d923f, f, start subscript, x, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0d923f, squared
  • 如果H, is greater than, 0, 函数肯定在该点left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis具有局部最大值/最小值.
    • 如果 start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, is greater than, 0, 这是个最小值。.
    • 如果 start color #0c7f99, f, start subscript, x, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99, is less than, 0, 这是个最大值。
  • 如果 H, is less than, 0 该功能肯定有一个鞍点为 left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis。
  • 如果 H, equals, 0, 我们没有足够信息去得出结果.

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