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 $\left(x_{0}, y_{0} \right)$ 的某邻域内有连续的二阶偏导数,且 $f_{x}^{\prime}\left(x_{0}, y_{0} \right)$ $=$ $0$, $f_{y}^{\prime}\left(x_{0}, y_{0} \right)$ $=$ $0$; 则极值判别公式 $AC$ $-$ $B^{2}$ 中的 $A$, $B$ 和 $C$ 各等于多少？

选项

[A].   $\begin{cases} & A = f_{y y}^{\prime \prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{x x}^{\prime \prime}\left(x_{0}, y_{0} \right) \end{cases}$

[B].   $\begin{cases} & A = f_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{y y}^{\prime \prime}\left(x_{0}, y_{0} \right) \end{cases}$

[C].   $\begin{cases} & A = f_{x}^{\prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{y}^{\prime}\left(x_{0}, y_{0} \right) \end{cases}$

[D].   $\begin{cases} & A = f_{y x}^{\prime \prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{y y}^{\prime \prime}\left(x_{0}, y_{0} \right) \end{cases}$

   答 案   

$\begin{cases} & A = f_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{y y}^{\prime \prime}\left(x_{0}, y_{0} \right) \end{cases}$