f(x,y)=k[sub]f[/sub] (3ℯ[sup]-(y + 1)² - x²[/sup] (x - 1)² - ℯ[sup]-(x + 1)² - 8²[/sup] / 3 + ℯ[sup]-x² - y²[/sup] (10x³ - 2x + 10y⁵))[size=85][br]*[i]The applet will work at a much faster speed if you download it to a desktop computer.[br][i] The order of operations for computing stationary points can be found in the [i][url=https://www.geogebra.org/m/te2vqwjq]applet[/url].[/i][/i][br][br] [b] Interactively find and compute local extrema of a nonlinear function of two variables without using its derivatives.[/b][br] Under the applet, you will find the stationary points of the function in question, calculated on a desktop computer, and you can compare them with more accurate calculations in CAS based on knowledge of the analytical formulas for its partial derivatives.[br] [/i]The iteration process consists of no+2 steps. When searching for local [b][color=#ff0000]maxima[/color][/b] and [b][color=#0000ff]minima[/color][/b], the function values ​​of the iteration points are sorted for all steps. However, such sorting is difficult because these values ​​coincide within the accuracy of the GeoGebra algebra. Therefore, to find the [b][color=#ff0000]largest[/color]/[color=#0000ff]smallest[/color][/b] value, this sorting is additionally performed in GeoGebra CAS with higher accuracy.[br] To calculate [b][color=#93c47d]saddle[/color][/b] points,[b] d[/b][sub][b]i[/b] [/sub]- the distance between two characteristic points for each iteration - was [i]chosen as a criterion[/i]. The iteration is considered optimal if this distance is [i]minimal[/i]. These distances for different iterations differ enough that the calculations can be done without using CAS.[br] *Explanations of the algorithms for searching for stationary points can be found in the applets [url=https://www.geogebra.org/m/ef6s3hyj]1[/url] and [url=https://www.geogebra.org/material/show/id/hcgdjdyf]2[/url].[/size]

[b]List of stationary points of the function f₄(x, y) found by CAS GeoGebra:[br][color=#ff0000][u][size=85]Local maximum points[/size][/u][/color]➟[br][size=85] [color=#cc0000]{(-0.4655263056553, -0.6217120029846), (1.285162657453, -0.004833795524588), (-0.01059996592404, 1.580343715217)}[/color];[/size][br][color=#0000ff][u][size=85]Local minimum points[/size][/u][/color]➟[br][size=85][color=#1e84cc] {(0.2288469494985, -1.626050253893), (0.3052469976531, 0.3238381792756), (-1.35971813141, 0.2144664990768)}[/color][color=#0000ff];[/color][/size][br][color=#38761d][u][size=85]Saddle points[/size][/u][/color]➟[size=85] [br][color=#6aa84f] {(0.4203044911699, -0.3877541612692), (1.097503671463, 0.8540090517851), (-0.2806862254121, 0.4785049750475)}}.[/color][/size][/b]

[size=85] Two independent methods compare the coordinates of possible local extrema (here -[color=#900000]Closes Points[/color]), function values, and the corresponding values of partial derivatives (which should be zero) at these points.[br] -[i]In the first method (1)[/i], the [url=https://www.geogebra.org/m/we7hu82b]calculation[/url] was performed using CAS GeoGebra based on the knowledge of the partial derivatives.[br]-[i]In the second method ([color=#ff00ff]2[/color]),[/i] the calculations are performed using my proposed algorithm: Descent-ascent numerical method for finding the stationary points of a function of two variables without using its derivatives.[br] In the tables, in the [color=#ff00ff]second row[/color], you can see the iteration number whose coordinates correspond to the local [color=#ff0000][b]maximum[/b][/color] or [color=#0000ff][b]minimum[/b][/color]. Due to the fact that the function values at points [b]no+2[/b] of the iterations are almost identical within the accuracy of the GeoGebra algebra. For this reason, the calculations were duplicated by sorting close function values using [b]CAS[/b], where the accuracy of the calculations is much higher.[br] To calculate [b]s[color=#38761d]addle points[/color][/b], d[sub]i[/sub] - the distance between two characteristic points for each iteration - was chosen as a criterion. The iteration is considered optimal if this distance is [b][i]minimal[/i][/b]. These distances for different iterations differ enough that the calculations can be done without using CAS.[/size]

[size=85]Obviously, the [color=#ff7700][i][b]relative error (fx,fy)/f[/b][/i][/color] of partial derivatives of the first CAS -method([b]1[/b]) is very small. The accuracy of the calculations using the proposed method ([color=#ff00ff][b]2[/b][/color]) for all types of local extrema is also quite high, as can be seen from the tables.[/size]