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

Critical Points of Function f(x,y): Local Max/Min/Saddle
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2019-04-28 17:43
CalculusDerivativeOptimization
This example demonstrates how to visualize and analyze the critical points of a multivariable function f(x,y) using GeoGebra, including local maxima, minima, and saddle points. Key math concepts involve multivariable functions, partial derivatives, Hessian matrix, and second derivative tests. GeoGebra commands like Simplify, ImplicitCurve, Zip, Sequence, TableText, LaTeX, Sort, Length, Max, RotateText are utilized, along with objects such as f, fx, fy, fxx, fxy, fyy, etc. Suitable for university-level calculus instruction or self-study on optimization techniques in multivariable calculus.

GeoGebra Commands & Parameters

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