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
- 若 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.
- 若 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
- 若 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=0∑nk!f(k)(x0)(x−x0)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)(x−x0)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 .
- 若 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=0∑nk!1[(x−x0)∂x∂+(y−y0)∂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[(x−x0)∂x∂+(y−y0)∂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=⎝⎜⎜⎜⎛x1x2⋮xn⎠⎟⎟⎟⎞,AAA=⎝⎜⎜⎜⎛a11a21⋮an1a12a22⋮an2……⋱…a1na2n⋮ann⎠⎟⎟⎟⎞
- 若 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 .
- 若 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 .
- 若 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 h→0 时,由极限的保号性不难发现
- 若 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.
- 若 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 有
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(h∂x∂+k∂y∂)2f(x0+θh,y0+θk)
其中 0 < θ < 1 0\lt \theta\lt 1 0<θ<1 .
令
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),
则
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)
令
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 h→0,k→0 时,由极限的保号性不难发现.
- 若 H H H 为正定矩阵,即 A > 0 , A C − B 2 > 0 A>0,AC-B^2>0 A>0,AC−B2>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.
- 若 H H H 为负定矩阵,即 A < 0 , A C − B 2 > 0 A<0,AC-B^2>0 A<0,AC−B2>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.
- 若 H H H 为不定矩阵,即 A C − B 2 < 0 AC-B^2<0 AC−B2<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)=⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛∂x12∂2f∂x2∂x1∂2f⋮∂xn∂x1∂2f∂x1∂x2∂2f∂x22∂2f⋮∂xn∂x2∂2f⋯⋯⋱⋯∂x1∂xn∂2f∂x2∂xn∂2f⋮∂xn2∂2f⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣P0
- 若 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.
- 若 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.
- 若 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.