极值充分条件

设二元函数\(f\)在点\(P_0(x_0,y_0)\)的某邻域\(U(P_0)\)上具有二阶连续偏导数，且\(P_0\)是\(f\)的稳定点。则当\(H_f(P_0)\)是正定矩阵时，\(f\)在点\(P_0\)处取得极小值；当\(H_f(P_0)\)是负定矩阵时，\(f\)在点\(P_0\)处取得极大值；当\(H_f(P_0)\)是不定矩阵，\(f\)在点\(P_0\)不取极值

证：

由\(f\)在\(P_0\)的二阶泰勒公式

\[\begin{align} f(x,y)-f(x_0,y_0)=&\\ &\nabla f(x_0,y_0)^T\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}-\frac12(\Delta x,\Delta y)H_f(P_0)\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}+o(\Delta x^2+\Delta y^2)\\ &&\\ =&(f_x,f_y)^T\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}-\frac12(\Delta x,\Delta y)H_f(P_0)\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}+o(\Delta x^2+\Delta y^2) \end{align} \]

假定\(f\)具有二阶连续偏导数，并记作：

\[H_f(P_0)=\begin{pmatrix}f_{xx}(P_0)&f_{xy}(P_0)\\f_{yx}(P_0)&f_{yy}(P_0) \end{pmatrix}=\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy} \end{pmatrix}_{P_0} \]

由于\(f\)具有二阶连续偏导数，所以\(f_{xy}=f_{yx}\)

由于\(P_0\)是\(f\)的稳定点，所以\(f_x(P_0)=f_y(P_0)=0\)，有

\[\begin{align}f(x,y)-f(x_0,y_0)=&\frac12(\Delta x,\Delta y)H_f(P_0)(\Delta x,\Delta y)^T+o(\Delta x^2+\Delta y^2)\\ =&\frac12(\Delta x,\Delta y)\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy} \end{pmatrix}_{P_0}(\Delta x,\Delta y)^T\\=& f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2) \end{align} \]

由二元一次方程\(ax^2+bx+c\)，不妨令\(a=f_{xx},\,b=2f_{xy}\Delta y,\, c=f_{yy}\Delta y^2\),则\(\Delta=b^2-4ac=4f_{xy}^2\Delta y^2-4f_{xx}f_{yy}\Delta y^2=f_{xy}^2-f_{xx}f_{yy}\)

1. \(f_{xx}>0\),\(\Delta=f_{xy}^2-f_{xx}f_{yy}<0\)，\(f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)\)是一个开口向上，与x轴没有交点的抛物线，此时\(f(x,y)-f(x_0,y_0)=f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)>0\)成立，得证\(f\)在\(P_0\)处取得极小值
2. \(f_{xx}<0,\,\Delta=f_{xy}^2-f_{xx}f_{yy}<0\)，则\(f\)为开口向下，与x轴没有交点的抛物线，函数恒小于0，此时\(f(x,y)-f(x_0,y_0)=f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)<0\)，即\(f\)在点\(P_0\)处取得极大值
3. \(\Delta=f_{xy}^2-f_{xx}f_{yy}>0\)时，抛物线与x轴有交点，有正有负，\(f\)在\(P_0\)处不能取得极值
4. \(\Delta=f_{xy}^2-f_{xx}f_{yy}=0\)时，不能肯定\(f\)是否在点\(P_0\)处取得极值

若\(H_f\)正定，则\(H_f\)的顺序主子式都大于0，所以\(f_{xx}>0,f_{xx}f_{yy}-f_{xy}^2>0\)时，恰好\(H_f\)正定，且\(f\)在\(P_0\)处取得极小值，

若\(H_f\)负定，\(f_{xx}<0,f_{xx}f_{yy}-f_{xy}^2>0\)时，\(f\)在\(P_0\)处取得极大值

若\(H_f\)不定，则不取得极值