f\left(x_0+\Delta x, y_0+\Delta y\right)=f\left(x_0, y_0\right)+f_x^{\prime}\left(x_0, y_0\right) \Delta x+f_y^{\prime}\left(x_0, y_0\right) \Delta y\\+\frac{1}{2}\left[f_{x x}^{\prime \prime}\left(x_0, y_0\right)(\Delta x)^2+2 f_{x y}^{\prime \prime}\left(x_0, y_0\right) \Delta x \Delta y+f_{y y}^{\prime \prime}\left(x_0, y_0\right)(\Delta y)^2\right]+\ldots \ldots
若 (x_0,y_0) 是极值点，该点是驻点，所以
f_x^{\prime}\left(x_0, y_0\right)=0\\ f_y^{\prime}\left(x_0, y_0\right)=0
令
h(\Delta x, \Delta y)=f_{x x}^{\prime \prime}\left(x_0, y_0\right)(\Delta x)^2+2 f_{x y}^{\prime \prime}\left(x_0, y_0\right) \Delta x \Delta y+f_{y y}^{\prime \prime}\left(x_0, y_0\right)(\Delta y)^2
若 h(\Delta x, \Delta y)>0 ，说明 (x_0,y_0) 周围的点比 f\left(x_0, y_0\right) 大，则 (x_0,y_0) 为极小值。 若 h(\Delta x, \Delta y)<0 ，说明 (x_0,y_0) 周围的点比 f\left(x_0, y_0\right) 小，则 (x_0,y_0) 为极大值。
h(\Delta x, \Delta y) 可以整理成二次方程的形式：
h(\Delta x, \Delta y)=\frac{1}{2}(\Delta y)^2\left[f_{x x}^{\prime \prime} \cdot\left(\frac{\Delta x}{\Delta y}\right)^2+2 \cdot f_{x y}^{\prime \prime}\left(\frac{\Delta x}{\Delta y}\right)+f_{y y}^{\prime}\right]
A=f_{x x}^{\prime \prime}\\ B=f_{x y}^{\prime \prime}\\ C=f_{yy}^{\prime \prime}
当 B^2-AC<0 且 A>0 时， h(\Delta x, \Delta y)>0 . 当 B^2-AC<0 且 A<0 时， h(\Delta x, \Delta y)<0 .
