极值存在的必要条件(B013)

问题

设 $z=f(x, y)$ 在点 $\left(x_{0}, y_{0}\right)$ 的一阶偏导数存在, 且 $\left(x_{0}, y_{0}\right)$ 是 $z=$ $f(x, y)$ 的极值点, 则可以推出以下哪个选项所示的结论?

选项

[A].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}\right)}$ $\neq$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}\right)}$

[B].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $1$, $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $1$

[C].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$, $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$

[D].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x, y\right)}$ $=$ $0$, $\left.\frac{\partial z}{\partial y}\right|_{\left(x, y \right)}$ $=$ $0$


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$\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$, $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$


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