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函数f(x, y)的临界点性质：局部极大/极小值与鞍点

函数f(x, y)的临界点性质：局部极大/极小值与鞍点
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2019-04-25 17:32
函数微积分导数
这个范例演示了如何通过解析几何方法研究函数f(x, y)的临界点性质，包括局部极大值、局部极小值和鞍点。核心数学概念涉及导数、矩阵和微积分。通过使用GeoGebra命令如ImplicitCurve、Simplify、Zip等，可以动态展示函数的性质。适合大学数学课程中的微积分和解析几何教学。

GeoGebra 命令及参数

  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