The function $f(x,y)=3x^2y+y^3-3x^2-3y^2+2$

1. has a saddle point at (-1,1)
2. has a minimum point at (-1,1)
3. has a maximum point at (0,2)
4. has four minimum points

$$f_x(x,y)=6x(y-1), \\ f_y(x,y)=3(x^2+y^2-2y)$$

$$f_{xx}(x,y)=f_{yy}(x,y)=6(y-1),\\ f_{xy}(x,y)=6x$$

The following observation are important to note.

1) Since, by Young's theorem, $f_{xy} =f_{yx}$, $f_{xy}\cdot f_{yx}=(f_{xy})^2$. step 3 may also be written
$f_{xx}\cdot f_{yy}-(f_{xy})^2>0$. (这里说明判别式中的二阶偏导可以交换求导顺序)

2) If $f_{xx}\cdot f_{yy}<(f_{xy})^2$, (a) when $f_{xx}$ and $f_{yy}$ have the same signs, the function is at an <u>infection point</u>; (b) when $f_{xx}$ and $f_{yy}$ have different signs, the function is at a <u>saddle point</u>, where the function is at a minimum when viewed from one axis, here the x axis, but at a maximum when viewed from the other axis, here the y axis. (这里就是值得商榷的地方)

3) If $f_{xx}\cdot f_{yy}=(f_{xy})^2$, the test is inconclusive.

whzecomjm
2018年7月11日

1. 实际上一维函数的拐点也可以称为一维鞍点.