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Critical Points of f(x,y): Local Max/Min/Saddle

Critical Points of f(x,y): Local Max/Min/Saddle
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2021-12-27 20:06
FunctionSolid GeometryCalculusDerivativeOptimization
In collections ₁ΛM 0 -Critical points and their nature using hessian matrix
This example demonstrates the visualization of critical points (local maxima, minima, and saddle points) of a function f(x,y) in 3D space. Key math concepts include multivariable functions, partial derivatives, Hessian matrix, and second derivative test. GeoGebra commands like ImplicitCurve, Zip, Sequence, TableText, LaTeX, Sort, Length, Max, RotateText are used for plotting implicit curves, handling sequences, displaying text/formulas, sorting, and rotating elements. Suitable for university-level calculus, multivariable function analysis, and teaching demonstrations.

GeoGebra Commands & Parameters

  1. ImplicitCurve(fx(x, y)) → ax
  2. ImplicitCurve(fy(x, y)) → ay
  3. Zip({{fxx(a), fxy(a)}, {fyx(a), fyy(a)}}, a, l1) → M
  4. Zip(Determinant[a], a, M) → D
  5. Zip(fxx(a), a, l1) → f_{xx}
  6. Sequence(If[D(i) > 0, If[f_{xx}(i) > 0, "local minimum", "local maximum"], If[(D(i) * f_{xx}(i)) ≟ 0, "?", "saddle point"]], i, 1, Length[l1]) → l4, {"saddle point", "local maximum", "local maximum", "saddle point", "saddle point"}
  7. Zip((x(a), y(a), f(a)), a, l1) → l5
  8. Zip(fx(a), a, l1) → lfx
  9. Zip(fy(a), a, l1) → lfy
  10. TableText(Append["i", 1…Length[l1]], Append["critical points", l5], Append["f_{x}", lfx], Append["f_{y}", lfy], Append["f_{xx}", f_{xx}], Append["Det", D], Append["", l4], "v") → text1
  11. LaTeX(f, true) → text2
  12. Sequence(If[D(i) > 0, If[f_{xx}(i) > 0, 1, 2], If[D(i) ≟ 0, -5, -1]], i, 1, Length[l1]) → l4a, {-1, 2, 2, -1, -1}
  13. Sort(l5, l4a) → l5a
  14. Sort(l1, l4a) → l1a
  15. Length(lmax) → lämax
  16. Length(lmin) → lämin
  17. Length(lsad) → läsad
  18. Sort(l4, l4a) → l4aa
  19. TableText(Append["i", 1…Length[l1]], Append["critical points", l5a], Append["", l4aa], "v") → text1a
  20. Length(lq) → läq
  21. ImplicitCurve(f(x, y) - z0) → IFxy
  22. Zip((x(a), y(a)), a, lsad) → lsad_2
  23. Zip((x(a), y(a)), a, lq) → lq_2
  24. Zip((x(a), y(a)), a, lmax) → lmax_2
  25. Zip((x(a), y(a)), a, lmin) → lmin_2
  26. Max(z(lmax)) → max
  27. Max(z(lmin)) → min
  28. RotateText(txt1, 90°) → txt2

AI Drawing Prompt Reference

Plot the function f(x,y) in 3D space, first draw the contour lines of f(x,y), then find the first and second partial derivatives, use the signs of the second partial derivatives to determine the nature of each critical point, and finally mark these critical points with their properties.
The prompt above was compiled from the original drawing commands and is for reference only.