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Nature of Critical Points of Function f(x, y): Local Max/Min and Saddle (1)

Nature of Critical Points of Function f(x, y): Local Max/Min and Saddle (1)
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2019-04-29 14:30
FunctionCalculusDerivativeOptimizationGraphing
This example demonstrates the nature of critical points of a function f(x, y), including local maxima, local minima, and saddle points. Key mathematical concepts include derivatives and second derivatives. Using commands like Simplify and ImplicitCurve, the construction shows the critical points of the function and uses text rotation to illustrate different scenarios. Suitable for university calculus courses.

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