GeoGebra

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

Author:Roman Chijner
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

2 and 3. ●Saddle points with "three" and "four" legs

2 and 3. ●Saddle points with "three" and "four" legs
In the case of saddle point "four" legs at the point (0,0,0), the function f(x,y) as well as its derivatives have an uncertainty - subject of discussion!

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Mappings for the cases of three stationary points. Localization of points on (a) contour map, (b) a graph of f(x,y), (c) a fragment, in a small domain. For a detailed discussion of Example 7 on the scheme for calculating stationary points of a function of two variables, see the link.

8.

8.
[size=85]Displaying images of the function f(x,y) and its fragment.[/size]
Displaying images of the function f(x,y) and its fragment.

f(x,y)=x y (x+y)(1+y)

f(x,y)=x y (x+y)(1+y)

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