1. Visualization of a numerical method for determining the type of extrema of functions with two variables on a contour map without using derivatives

[size=85]For the function f(x,y) with two variables on a contour map, a numerical method is proposed to determine the nature of the extrema without using the derivatives.[br] The analysis is based on the composite function Δf(α) - changes of the function f(x,y) for the corresponding points on the circles (r; α) and (r+Δr; α). The applet determines areas of monotonic increase or decrease for the analyzed function f(x,y) on a test circle that is described around a critical point on the contour map. [br] ✱If [color=#9900ff]Δf(α)<0[/color] for α∈[0,2 π ], then f(x,y) [color=#0000ff]decreases[/color] at the ends of the radii r and r+Δr of the circles (for each α), i.e. there is a [color=#ff0000]local maximum[/color] at their center.[br] ✱If [color=#9900ff]Δf(α)>0[/color] for α∈[0,2 π ], then f(x,y) [color=#ff0000]increases[/color] at the ends of the radii r and r+Δr of the circles (for each α), i.e. there is a [color=#0000ff]local minimum[/color] at their center.[br] ✱If [color=#9900ff]Δf(α)[/color] for [color=#9900ff]α∈[0,2 π ][/color], is an alternating function with zeros (in the color "[color=#1e84cc]Deep Sky Blue[/color]"), then f(x,y) has [color=#ff0000]increasing[/color] and [color=#0000ff]decreasing[/color] function intercepts at the edges of these circles, i.e. there is a [color=#00ff00]saddle point [/color]in the center of these circles. Near this point, the surface has the shape of a saddle around the critical point: - [i]concave upwards [/i]in one direction, - [i]concave downwards[/i] in another direction. 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[/img][br][sup]*[/sup]The applet offers the possibility to check the accuracy of these calculations by tracing (button - "Trace On") the calculated monotonicity limits.[br][sup]**[/sup]In case index=1 it is possible to enter functions from the input box.[br][/size][sup][/sup][size=85][sup]***[/sup]Further improvements of the described method can be found in the [url=https://www.geogebra.org/m/v6pwykyh]applet[/url]. [/size]
[size=85]A detailed solution of the problem for the case index=10 according to the above diagram is discussed in detail in the [url=https://socratic.org/questions/what-are-the-extrema-and-saddle-points-of-f-x-y-f-x-y-xy-1-x-y]Calculus[/url] "What are the extrema and saddle points of f(x,y)=xy(1-x-y)?[/size]
Function graphs (index=3) with the same contour plot: f(x,y)=(x²-y²)ᴷ for k=1 and k=2
[size=85]1. Concentric closed contour lines always indicate either a [color=#0000ff]local minimum[/color] or a local maximum. [br] Critical points that are not a local maximum or minimum are usually [color=#00ff00]saddle points[/color]. It is a crossing of two contour lines of f(x,y). The surface around the critical point has the shape of a saddle: - [i]concave upwards[/i] in one direction, - [i]concave downwards[/i] in another.[br]2. If a contour line intersects itself, the point could be a[br]★ [color=#00ff00]Saddle point[/color],[br]★ [color=#0000ff]local minimum[/color], or[br]★ [color=#ff0000]local maximum[/color].[br] Here is a pair of function graphs with the same contour plot. The figure considers two cases for index=3:[br] (a) for k=1 -Hyperbolic paraboloid: z=x[sup]2[/sup]-y[sup]2[/sup] and [br] (b) for k=2 -z=(x[sup]2[/sup]-y[sup]2[/sup])[sup]2[/sup].[br] For a discussion, see the article "[url=https://math.stackexchange.com/questions/3345225/how-to-read-contour-plot]How to read contour plot[/url]?".[br]A similar case occurs for index=1: z=x y and z=(x y)[sup]2[/sup]. You can see this case for yourself by setting the [color=#00ff00]saddle point[/color] to a [color=#ff0000]local maximum[/color]. [/size]
Example (index=7): f(x,y)=x⁶+y⁶-15(x²+y²)
[size=85]In this example, there is [b]one[/b] [color=#ff0000]local maximum[/color] point, [b]three[/b] [color=#0000ff]local minimum points[/color], and [b]three[/b] [color=#00ff00]saddle points[/color].[/size]
Example (index=8): f(x,y)=3(1-x²) exp(-(x-0.5)²-(y+1)²)-2(0.2x-x³-y⁵)exp(-x²-y²)sin(x-y)
Example (index=9): Is N a saddle point?
[size=85]Is [color=#00ff00]N [/color]a [color=#00ff00]saddle point[/color] here?[br][url=https://www.wolframalpha.com/input?i=%28x%5E2%2B+y%5E2%29%5E2%2B3+x%5E2+y-+y%5E3+]https://www.wolframalpha.com/input?i=%28x%5E2%2B+y%5E2%29%5E2%2B3+x%5E2+y-+y%5E3+[/url][br] f(x, y) = (x² + y²)² + 3x² y - y³;[br] f[sub]x[/sub] (x,y)=4x³+2xy(2y+3);[br] f[sub]y[/sub](x,y)=x²(4y+3)+y²(4y-3); [br] f[sub]xx[/sub](x,y)=12x² + 4y² + 6 y;[br] f[sub]xy[/sub](x,y)=8 xy+6 x;[br] f[sub]yx[/sub](x,y)=8 xy+6 x;[br] f[sub]yy[/sub](x,y)=4x²+12y²-6y;[br] D(x,y)=|f[sub]xx[/sub]*f[sub]yy[/sub]-f[sub]xy[/sub]*f[sub]yx[/sub]|;[br]D(x,y)= (12x² + 4y² + 6y) (4x² + 12y² - 6y) - (8x y + 6x)² =[br]=12 (4x⁴ + 8x² y² - 12x² y - 3x² + 4y⁴ + 4y³ - 3y²);[br]⇒f[sub]x[/sub](0,0)=0; f[sub]y[/sub](0,0)=0 ; D(0,0)=0[br]Regarding the case D=0 in the literature we have ambiguous arguments:[br][url=https://en.wikipedia.org/wiki/Second_partial_derivative_test]https://en.wikipedia.org/wiki/Second_partial_derivative_test[/url][br]... critical point (0, 0) the second derivative test is insufficient, and [i]one must use higher order tests or other tools to determine the behavior of the function[/i] at this point. ([b]In fact[/b], one can show that f takes both [color=#ff0000]positive [/color]and [color=#0000ff]negative[/color] values in small neighborhoods around (0, 0) and so [b]this point is a [color=#00ff00]saddle point[/color] of f.[/b])[/size]
Scheme for the calculation of the stationary points of the 2-variable function
[size=85]From [url=https://ayraethazide.tumblr.com/post/74061500099/here-are-some-notes-on-the-classification-of]https://ayraethazide.tumblr.com/post/74061500099/here-are-some-notes-on-the-classification-of[br][/url] A detailed solution of the problem for the case index=10 according to the above diagram is discussed in detail in the [url=https://socratic.org/questions/what-are-the-extrema-and-saddle-points-of-f-x-y-f-x-y-xy-1-x-y]Calculus[/url] "What are the extrema and saddle points of f(x,y)=xy(1-x-y)?[/size]

Information: 1. Visualization of a numerical method for determining the type of extrema of functions with two variables on a contour map without using derivatives