函数极值问题

本文旨在加深对二元函数极值充分条件的理解,不追求完美证明.

文章目录

〇、前置知识

极值点的定义

  • P 0 P_0 P0 处的函数值大于 U ˚ ( P 0 ) \mathring{U}(P_0) U˚(P0) 内的函数值,则 P 0 P_0 P0 为极大值点.
  • P 0 P_0 P0 处的函数值小于 U ˚ ( P 0 ) \mathring{U}(P_0) U˚(P0) 内的函数值,则 P 0 P_0 P0 为极小值点.

泰勒中值定理

  • f ( x ) f(x) f(x)​​​​​ 在 x 0 x_0 x0​​​​​ 某领域内 ( n + 1 ) (n+1) (n+1)​​​​​ 阶可导,则 f ( x ) = ∑ k = 0 n f ( k ) ( x 0 ) k ! ( x − x 0 ) k + R n ( x ) , f(x)=\overset{n}{\underset{k=0}{\sum}}\dfrac{f^{(k)}(x_0)}{k!}(x-x_0)^k+R_n(x), f(x)=k=0nk!f(k)(x0)(xx0)k+Rn(x),​​​​​ 其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 R_n(x)=\dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} Rn(x)=(n+1)!f(n+1)(ξ)(xx0)n+1​​​​​ 为 L a g r a n g e Lagrange Lagrange​​​​​ 余项, ξ \xi ξ​​​ 介于 x 0 , x x_0,x x0,x​​​ 之间 .
  • f ( x , y ) f(x,y) f(x,y)​​ 在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​ 某领域内有 ( n + 1 ) (n+1) (n+1)​​ 阶连续偏导数,则 f ( x , y ) = ∑ k = 0 n 1 k ! [ ( x − x 0 ) ∂ ∂ x + ( y − y 0 ) ∂ ∂ y ] k f ( x 0 , y 0 ) + R n , f(x,y)=\overset{n}{\underset{k=0}{\sum}}\dfrac{1}{k!}\bigg[(x-x_0)\dfrac{\partial}{\partial x}+(y-y_0)\dfrac{\partial}{\partial y}\bigg]^kf(x_0,y_0)+R_n, f(x,y)=k=0nk!1[(xx0)x+(yy0)y]kf(x0,y0)+Rn,​​ R n = 1 ( n + 1 ) ! [ ( x − x 0 ) ∂ ∂ x + ( y − y 0 ) ∂ ∂ y ] n + 1 f ( ξ , η ) , R_n=\dfrac{1}{(n+1)!}\bigg[(x-x_0)\dfrac{\partial}{\partial x}+(y-y_0)\dfrac{\partial}{\partial y}\bigg]^{n+1}f(\xi,\eta), Rn=(n+1)!1[(xx0)x+(yy0)y]n+1f(ξ,η),​​ ξ \xi ξ 介于 x 0 , x x_0,x x0,x 之间, η \eta η 介于 y 0 , y y_0,y y0,y​ 之间.

二次型

设二次型 f ( x ) = x T A x f(\pmb{x})=\pmb{x}^T\pmb{A}\pmb{x} f(xxx)=xxxTAAAxxx ,其中
x = ( x 1 x 2 ⋮ x n ) , A = ( a 11 a 12 … a 1 n a 21 a 22 … a 2 n ⋮ ⋮ ⋱ ⋮ a n 1 a n 2 … a n n ) \pmb{x}=\begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{pmatrix},\pmb{A}=\begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix} xxx=x1x2xn,AAA=a11a21an1a12a22an2a1na2nann

  • A A A​​ 为正定矩阵(特征值全为正),则 x ≠ 0 \pmb x \ne 0 xxx=0​ 时恒有 f ( x ) > 0 f(\pmb x)\gt 0 f(xxx)>0​​​​ .
  • A A A​ 为负定矩阵(特征值全为负),则 x ≠ 0 \pmb x \ne 0 xxx=0​ 时恒有 f ( x ) < 0 f(\pmb x)\lt 0 f(xxx)<0​ .
  • A A A​ 为不定矩阵(特征值既有正又有负),则 x ≠ 0 \pmb x \ne 0 xxx=0​ 时 f ( x ) f(\pmb x) f(xxx)​ 有正有负.
  • 否则 f ( x ) f(\pmb x) f(xxx) 取值的正负性需进一步判断.

正定矩阵的判别:各阶顺序主子式均为正.

负定矩阵的判别:奇数阶顺序主子式为负,偶数阶顺序主子式为正.

一、一元函数的极值

设一元函数 f ( x ) f(x) f(x)​ 在 x 0 x_0 x0​ 处具有二阶导数,且 f ′ ( x 0 ) = 0 f'(x_0)=0 f(x0)=0​ .

由泰勒中值定理及一阶导数为 0 0 0​ 有
f ( x 0 + h ) = f ( x 0 ) + f ′ ′ ( x 0 + θ h ) 2 ! h 2 , f(x_0+h)=f(x_0)+\dfrac{f”\left(x_0+ \theta h \right) }{2!}h^{2}, f(x0+h)=f(x0)+2!f(x0+θh)h2,
其中 0 < θ < 1 0\lt \theta\lt 1 0<θ<1 .

h → 0 h\to 0 h0 时,由极限的保号性不难发现

  • f ′ ′ ( x 0 ) > 0 f”\left( x_0 \right)\gt0 f(x0)>0​​​​​​ ,则 x 0 x_0 x0​​​​​​ ​邻域内的函数值大于 f ( x 0 ) f(x_0) f(x0) ,函数在 x 0 x_0 x0​​​​​​​ 处取得极小值.
  • f ′ ′ ( x 0 ) < 0 f”\left( x_0 \right)\lt 0 f(x0)<0 ,则 x 0 x_0 x0 邻域内的函数值小于 f ( x 0 ) f(x_0) f(x0) ,函数在 x 0 x_0 x0​ 处取得极大值.
  • 否则需进一步讨论以确定 x 0 x_0 x0 处函数值与其附近的大小关系.

二、二元函数的极值

设二元函数 f ( x , y ) f(x,y) f(x,y)​​​ 在点 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​​ 处具有二阶连续偏导数,且 f x ( x 0 , y 0 ) = f y ( x 0 , y 0 ) = 0 f_x(x_0,y_0)=f_y(x_0,y_0)=0 fx(x0,y0)=fy(x0,y0)=0​​ .

由泰勒中值定理及一阶偏导数为 0 0 0
f ( x 0 + h , y 0 + k ) = f ( x 0 , y 0 ) + 1 2 ! ( h ∂ ∂ x + k ∂ ∂ y ) 2 f ( x 0 + θ h , y 0 + θ k ) f(x_0+h,y_0+k) =f(x_0,y_0)+\dfrac{1}{2!}\left( h\dfrac{\partial }{\partial x}+k\dfrac{\partial }{\partial y}\right) ^{2}f\left( x_{0}+\theta h,y_{0}+\theta k \right) f(x0+h,y0+k)=f(x0,y0)+2!1(hx+ky)2f(x0+θh,y0+θk)
其中 0 < θ < 1 0\lt \theta\lt 1 0<θ<1 .


A θ = f x x ( x 0 + θ h , y 0 + θ k ) , B θ = f x y ( x 0 + θ h , y 0 + θ k ) = f y x ( x 0 + θ h , y 0 + θ k ) , C θ = f y y ( x 0 + θ h , y 0 + θ k ) , \begin{aligned} &A_{\theta}=f_{xx}(x_0+\theta h,y_0+\theta k),\\ &B_{\theta}=f_{xy}(x_0+\theta h,y_0+\theta k)=f_{yx}(x_0+\theta h,y_0+\theta k),\\ &C_{\theta}=f_{yy}(x_0+\theta h,y_0+\theta k), \end{aligned} Aθ=fxx(x0+θh,y0+θk),Bθ=fxy(x0+θh,y0+θk)=fyx(x0+θh,y0+θk),Cθ=fyy(x0+θh,y0+θk),

f ( x 0 + h , y 0 + k ) = f ( x 0 , y 0 ) + 1 2 [ A θ h 2 + B θ h k + B θ k h + C θ k 2 ] = f ( x 0 , y 0 ) + 1 2 ( h k ) ( A θ B θ B θ C θ ) ( h k ) \begin{aligned} f(x_0+h,y_0+k) &=f(x_0,y_0)+\dfrac{1}{2}\left[ A_{\theta}h^{2}+B_{\theta}hk+B_{\theta}kh+C_{\theta}k^{2}\right]\\ &=f(x_0,y_0)+ \dfrac{1}{2}\begin{pmatrix} h & k \end{pmatrix}\begin{pmatrix} A_{\theta } & B_{\theta } \\ B_{\theta } & C_{\theta } \end{pmatrix}\begin{pmatrix} h \\ k \end{pmatrix} \end{aligned} f(x0+h,y0+k)=f(x0,y0)+21[Aθh2+Bθhk+Bθkh+Cθk2]=f(x0,y0)+21(hk)(AθBθBθCθ)(hk)

H = ( A B B C ) = ( f x x f x y f y x f y y ) ∣ ( x 0 , y 0 ) H=\begin{pmatrix} A & B \\ B & C \end{pmatrix}= \left. \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}\right|_{(x_{0},y_{0})} H=(ABBC)=(fxxfyxfxyfyy)(x0,y0)
h → 0 , k → 0 h \to 0,k\to 0 h0,k0​ 时,由极限的保号性不难发现.

  • H H H​​​ 为正定矩阵,即 A > 0 , A C − B 2 > 0 A>0,AC-B^2>0 A>0,ACB2>0​​​,则 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​​ 领域内的函数值大于 f ( x 0 , y 0 ) f(x_0,y_0) f(x0,y0)​​​​ ,函数在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​​​ 处取得极小值.
  • H H H​​ 为负定矩阵,即 A < 0 , A C − B 2 > 0 A<0,AC-B^2>0 A<0,ACB2>0​​,则 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​ 领域内的函数值小于 f ( x 0 , y 0 ) f(x_0,y_0) f(x0,y0)​​ ,函数在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​ 处取得极大值.
  • H H H​​​ 为不定矩阵,即 A C − B 2 < 0 AC-B^2<0 ACB2<0​ ,则 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​​ 领域内既有函数值大于 f ( x 0 , y 0 ) f(x_0,y_0) f(x0,y0)​ 的点,又有函数值小于 f ( x 0 , y 0 ) f(x_0,y_0) f(x0,y0)​ 的点,于是函数在 ( x 0 , y 0 ) (x_0,y_0) (x0,y0)​ 处无极值.
  • 否则需进一步讨论以确定 ( x 0 , y 0 ) (x_0,y_0) (x0,y0) 处函数值与其附近的大小关系.

三、多元函数的极值

根据二元函数的极值充分条件,不难推广到多元函数的情况.

设多元函数 f ( x 1 , x 2 , ⋯   , x n ) f(x_1,x_2,\cdots,x_n) f(x1,x2,,xn)​​ 在点 P 0 ( a 1 , a 2 , ⋯   , a n ) P_0(a_1,a_2,\cdots,a_n) P0(a1,a2,,an)​​ 处具有二阶连续偏导数,且 f x 1 = f x 2 = ⋯ = f x n = 0 f_{x_1}=f_{x_2}=\cdots =f_{x_n}=0 fx1=fx2==fxn=0​​ .

构造 P 0 P_0 P0 黑塞矩阵( H e s s i a n   M a t r i x Hessian\ Matrix Hessian Matrix
H ( P 0 ) = ( ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ) ∣ P 0 H(P_0)= \left. \begin{pmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\partial x_n}\\ \dfrac{\partial^2 f}{\partial x_2\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \dfrac{\partial^2 f}{\partial x_n\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}\\ \end{pmatrix} \right|_{P_0} H(P0)=x122fx2x12fxnx12fx1x22fx222fxnx22fx1xn2fx2xn2fxn22fP0

  • H ( P 0 ) H(P_0) H(P0)​​​​​​​​ 为正定矩阵,则 P 0 P_0 P0​​​​​​​​ 领域内的函数值大于 f ( P 0 ) f(P_0) f(P0)​​​​​​​​​ ,函数在 P 0 P_0 P0​​​​​​​​​ 处取得极小值.
  • H ( P 0 ) H(P_0) H(P0)​​​​ 为负定矩阵,则 P 0 P_0 P0​​​​ 领域内的函数值小于 f ( P 0 ) f(P_0) f(P0)​​​​ ,函数在 P 0 P_0 P0​​​​ 处取得极大值.
  • H ( P 0 ) H(P_0) H(P0)​​​​​ 为不定矩阵 ,则 P 0 P_0 P0​​​​​ 领域内既有函数值大于 f ( P 0 ) f(P_0) f(P0)​​​​​ 的点,又有函数值小于 f ( P 0 ) f(P_0) f(P0)​​​​​ 的点,于是函数在 P 0 P_0 P0​​​​​ 处无极值.
  • 否则需进一步讨论以确定 P 0 P_0 P0​ 处函数值与其附近的大小关系.
    原文作者:_zom_bie
    原文地址: https://blog.csdn.net/qq_44729222/article/details/119891453
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