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

Nature of Critical Points of Function f(x, y): Local Max/Min and Saddle
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2019-04-25 17:32
FunctionCalculusDerivative
In collections ₁ΛM 0 -Critical points and their nature using hessian matrix
This example demonstrates the nature of critical points of a function f(x, y), including local maxima, minima, and saddle points, using geometric methods. Key mathematical concepts include derivatives, matrices, and calculus. GeoGebra commands such as ImplicitCurve, Simplify, and Zip are used to dynamically illustrate the function's properties. Suitable for university-level calculus and coordinate geometry courses.

GeoGebra Commands & Parameters

  1. ImplicitCurve(fy0(x, y)) → ay0
  2. ImplicitCurve(fx0(x, y)) → ax0
  3. Simplify(Derivative[f, x, 1]) → fx
  4. Simplify(Derivative[f, y, 1]) → fy
  5. ImplicitCurve(fx(x, y)) → ax
  6. ImplicitCurve(fy(x, y)) → ay
  7. Simplify(Derivative[fx, x, 1]) → fxx
  8. Simplify(Derivative[fx, y, 1]) → fxy
  9. Simplify(Derivative[fy, y, 1]) → fyy
  10. Simplify(Derivative[fy, x, 1]) → fyx
  11. Zip({{fxx(a), fxy(a)}, {fyx(a), fyy(a)}}, a, l1) → M
  12. Zip(Determinant[a], a, M) → D
  13. Zip(fxx(a), a, l1) → f_{xx}
  14. 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
  15. Zip((x(a), y(a), f(a)), a, l1) → l5
  16. Zip(fx(a), a, l1) → lfx
  17. Zip(fy(a), a, l1) → lfy
  18. 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
  19. LaTeX(f, true) → text2
  20. Sequence(If[D(i) > 0, If[f_{xx}(i) > 0, 1, 2], If[D(i) ≟ 0, -5, -1]], i, 1, Length[l1]) → l4a
  21. Sort(l5, l4a) → l5a
  22. Sort(l1, l4a) → l1a
  23. Length(lmax) → lämax
  24. Length(lmin) → lämin
  25. Length(lsad) → läsad
  26. Sort(l4, l4a) → l4aa
  27. TableText(Append["i", 1…Length[l1]], Append["critical points", l5a], Append["", l4aa], "v") → text1a
  28. Length(lq) → läq
  29. ImplicitCurve(f(x, y) - z0) → IFxy
  30. Zip((x(a), y(a)), a, lsad) → lsad_2
  31. Zip((x(a), y(a)), a, lq) → lq_2
  32. Zip((x(a), y(a)), a, lmax) → lmax_2
  33. Zip((x(a), y(a)), a, lmin) → lmin_2
  34. RotateText(txt1, 90°) → txt2