極值充分條件
設二元函數\(f\)在點\(P_0(x_0,y_0)\)的某鄰域\(U(P_0)\)上具有二階連續偏導數，且\(P_0\)是\(f\)的穩定點。則當\(H_f(P_0)\)是正定矩陣時，\(f\)在點\(P_0\)處取得極小值；當\(H_f(P_0)\)是負定矩陣時，\(f\)在點\(P_0\)處取得極大值；當\(H_f(P_0)\)是不定矩陣，\(f\)在點\(P_0\)不取極值
證：
由\(f\)在\(P_0\)的二階泰勒公式
\[\begin{align} f(x,y)-f(x_0,y_0)=&\\ &\nabla f(x_0,y_0)^T\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}-\frac12(\Delta x,\Delta y)H_f(P_0)\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}+o(\Delta x^2+\Delta y^2)\\ &&\\ =&(f_x,f_y)^T\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}-\frac12(\Delta x,\Delta y)H_f(P_0)\begin{pmatrix}\Delta x\\\ \Delta y \end{pmatrix}+o(\Delta x^2+\Delta y^2) \end{align} \]
假定\(f\)具有二階連續偏導數，並記作：
\[H_f(P_0)=\begin{pmatrix}f_{xx}(P_0)&f_{xy}(P_0)\\f_{yx}(P_0)&f_{yy}(P_0) \end{pmatrix}=\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy} \end{pmatrix}_{P_0} \]
由於\(f\)具有二階連續偏導數，所以\(f_{xy}=f_{yx}\)
由於\(P_0\)是\(f\)的穩定點，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\(a=f_{xx},\,b=2f_{xy}\Delta y,\, c=f_{yy}\Delta y^2\),則\(\Delta=b^2-4ac=4f_{xy}^2\Delta y^2-4f_{xx}f_{yy}\Delta y^2=f_{xy}^2-f_{xx}f_{yy}\)
- \(f_{xx}>0\),\(\Delta=f_{xy}^2-f_{xx}f_{yy}<0\)，\(f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)\)是一個開口向上，與x軸沒有交點的拋物線，此時\(f(x,y)-f(x_0,y_0)=f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)>0\)成立，得證\(f\)在\(P_0\)處取得極小值
- \(f_{xx}<0,\,\Delta=f_{xy}^2-f_{xx}f_{yy}<0\)，則\(f\)為開口向下，與x軸沒有交點的拋物線，函數恆小於0，此時\(f(x,y)-f(x_0,y_0)=f_{xx}\Delta x^2+2f_{xy}\Delta x\Delta y+f_{yy}\Delta y^2+o(\Delta x^2+\Delta y^2)<0\)，即\(f\)在點\(P_0\)處取得極大值
- \(\Delta=f_{xy}^2-f_{xx}f_{yy}>0\)時，拋物線與x軸有交點，有正有負，\(f\)在\(P_0\)處不能取得極值
- \(\Delta=f_{xy}^2-f_{xx}f_{yy}=0\)時，不能肯定\(f\)是否在點\(P_0\)處取得極值
若\(H_f\)正定，則\(H_f\)的順序主子式都大於0，所以\(f_{xx}>0,f_{xx}f_{yy}-f_{xy}^2>0\)時，恰好\(H_f\)正定，且\(f\)在\(P_0\)處取得極小值，
若\(H_f\)負定，\(f_{xx}<0,f_{xx}f_{yy}-f_{xy}^2>0\)時，\(f\)在\(P_0\)處取得極大值
若\(H_f\)不定，則不取得極值