Some pictures of visualization of a numerical method for determining the type of local extrema of functions with two variables on a contour map without using derivatives
Here are explanatory images from the applet.
Concentric closed contour lines always indicate either a local minimum or a local maximum. If a contour line intersects itself, the point could be a saddle point, local minimum, or local maximum.
1. ●Cross-shaped contour lines for both the saddle point, known as the monkey saddle, and the local minimum
![1. ●Cross-shaped contour lines for both the saddle point, known as the monkey saddle, and the local minimum](https://stage.geogebra.org/resource/ryjeymwr/sZXArEh0vefDSvlf/material-ryjeymwr.png)
2 and 3. ●Saddle points with "three" and "four" legs
![2 and 3. ●Saddle points with "three" and "four" legs](https://stage.geogebra.org/resource/xfy4whpn/cNENdCOjy1pcDwu9/material-xfy4whpn.png)
4.
![4.](https://stage.geogebra.org/resource/fchqrd8f/jgNXNTHApeFFaD6X/material-fchqrd8f.png)
5.
![5.](https://stage.geogebra.org/resource/rp6gjwaq/XpFRc0CBZW1OwUiC/material-rp6gjwaq.png)
6.
![6.](https://stage.geogebra.org/resource/xzkehdn4/jpK3qu5HGE940AIl/material-xzkehdn4.png)
7.
![7.](https://stage.geogebra.org/resource/nrbfqbfb/KLrl5Loz7BjcSymk/material-nrbfqbfb.png)
8.
![8.](https://stage.geogebra.org/resource/f3bpcpnr/qx3ms5vr9E9XHWj6/material-f3bpcpnr.png)
![[size=85]Displaying images of the function f(x,y) and its fragment.[/size]](https://stage.geogebra.org/resource/cve7xhh6/0uIYrKWPHIxEqF7n/material-cve7xhh6.png)
f(x,y)=x y (x+y)(1+y)
![f(x,y)=x y (x+y)(1+y)](https://stage.geogebra.org/resource/sy3aaf74/g6aOLftUoqGnXArH/material-sy3aaf74.png)