# 通过二元复合函数判断一元函数的极值点条件

## 二、解析

$$F(x, y)=0 \Rightarrow$$

$$[F(x, y)]_{x}^{\prime}=0 \Rightarrow$$

$$F_{x}^{\prime}+y_{x}^{\prime} F_{y}^{\prime}=0 \Rightarrow$$

$$y_{x}^{\prime}=\frac{-F_{x}^{\prime}}{F_{y}^{\prime}} \Rightarrow$$

$$F_{x}^{\prime}\left(x_{0}, y_{0}\right)=0, \quad F_{y}^{\prime}\left(x_{0}, y_{0}\right)>0 \Rightarrow$$

$$y_{x}^{\prime}\left(x_{0}, y_{0}\right)=0$$

$${\left[y_{x}^{\prime}\right]_{x}^{\prime}=\left[\frac{-F_{x}^{\prime}}{F_{y}^{\prime}}\right]_{x}^{\prime} \Rightarrow}$$

$$y_{x x}^{\prime \prime}=\frac{-\left(F_{x x}^{\prime \prime}+y_{x}^{\prime} F_{x y}^{\prime \prime}\right) F_{y}^{\prime}+F_{x}^{\prime}\left(F_{y x}^{\prime \prime}+y_{x}^{\prime} F_{y y}^{\prime \prime}\right)}{\left(F_{y}^{\prime}\right)^{2}}$$

$$x=x_{0}, \quad y=y_{0} \Rightarrow$$

$$\begin{cases} & y_{x}^{\prime}\left(x_{0}, y_{0}\right)=0 \\ & F^{\prime} _{x}\left(x_{0}, y_{0}\right)=0 \end{cases} \Rightarrow$$

$$y_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right) = \frac{-\left(F_{x x}^{\prime \prime} F_{y}^{\prime}\right)^{\prime}}{\left(F_{y}^{\prime}\right)^{2}}$$

$$y_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right)>0, \quad F_{y}^{\prime}\left(x_{0}, y_{0}\right)>0 \Rightarrow$$

$$F_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right)>0$$