极值存在的充分条件：判别公式中的 $A$, $B$, $C$ 都是多少？（B013）

问题

若已知函数 $z$ $=$ $f(x,y)$ 在点 $(x_0,y_0)$ 的某邻域内有连续的二阶偏导数,且 $f_x^{\prime }(x_0,y_0)$ $=$ $0$, $f_y^{\prime }(x_0,y_0)$ $=$ $0$; 则极值判别公式 $AC$ $-$ $B^2$ 中的 $A$, $B$ 和 $C$ 各等于多少？

选项

[A].   $\{\begin{array}{ll}& A=f_{yx}^{\prime \prime }(x_0,y_0)\\ & B=f_{xy}^{\prime \prime }(x_0,y_0)\\ & C=f_{yy}^{\prime \prime }(x_0,y_0)\end{array}$

[B].   $\{\begin{array}{ll}& A=f_{yy}^{\prime \prime }(x_0,y_0)\\ & B=f_{xy}^{\prime \prime }(x_0,y_0)\\ & C=f_{xx}^{\prime \prime }(x_0,y_0)\end{array}$

[C].   $\{\begin{array}{ll}& A=f_{xx}^{\prime \prime }(x_0,y_0)\\ & B=f_{xy}^{\prime \prime }(x_0,y_0)\\ & C=f_{yy}^{\prime \prime }(x_0,y_0)\end{array}$

[D].   $\{\begin{array}{ll}& A=f_x^{\prime }(x_0,y_0)\\ & B=f_{xy}^{\prime \prime }(x_0,y_0)\\ & C=f_y^{\prime }(x_0,y_0)\end{array}$

   答 案   

$\{\begin{array}{ll}& A=f_{xx}^{\prime \prime }(x_0,y_0)\\ & B=f_{xy}^{\prime \prime }(x_0,y_0)\\ & C=f_{yy}^{\prime \prime }(x_0,y_0)\end{array}$