Interactively find and compute local extrema of a nonlinear function of two variables
The purpose of this applet is to find and calculate the possible extremes of the nonlinear function f(x,y). The possibilities of such calculations are shown by the example of the sum of several nonlinear functions.
Method. The problem comes down to finding the first partial derivatives fx(x,y) and fy(x,y). Using equations with implicit functions fx(x,y)=0, fy(x,y)=0 in the CAS section of GeoGebra, the intersection points of these implicit functions are found numerically, which are possible extreme points.
The applet provides an approximate method to determine the type of these points: maximum, minimum, saddle. There is a test circle with radius p0 and three moving points arranged at 120° to each other. If the test point is red, then it is the maximum point and the points on the circle are blue, i.e. the function is decreasing ↘ from the test point A. If it is blue, then it is the minimum, green is a saddle point. This is an approximate method. To be sure, you can rotate these points in a circle and see how their color changes. When refining the result, you can reduce the value of the radius p0.
What actions can you do with this applet?
In section 1. Collect all extreme points in one list. To do this: Set the test point A at each intersection of the curves fx(x,y)=0 and fy(x,y)=0 and press the "click script" button one after the other and with the help of the Solve command a list of all possible local extrema will be formed.
In Section 2. You can see these extrema and their ordinal numbers, their number -Length[Solve].
In Section 3. With the help of the available script these points can be sorted into lists: Max, Min, Sad.
1. Visualizing a function of two variables using contour lines
![[size=85]Big doubts about the extremum h=13. The contour lines near it are the same as for the saddle point. Although there are examples where this pattern can be the same for max/min points. The problem is that the function at this point is ≈0. With the help of the scale factor it can be increased. But all these actions, it seems to me, are at the limit of accuracy of numerical calculations.[/size]](https://www.geogebra.org/resource/uucmvzu5/isLCBwKDAxrWuDtw/material-uucmvzu5.png)
2. Test point A to determine extremum type
3. Compare the work of commands: Solutions and Solve
