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函数 f(x,y) 的临界点性质：局部最大/最小/鞍点

函数 f(x,y) 的临界点性质：局部最大/最小/鞍点
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2019-04-28 17:43
微积分导数最值优化
①本范例演示了如何通过 GeoGebra 绘制并分析函数 f(x,y) 的临界点，包括局部最大值、最小值和鞍点；②涉及的核心数学概念有：多元函数、偏导数、Hessian 矩阵、二阶导数检验法；③作图要点：使用了 Simplify, ImplicitCurve, Zip, Sequence, TableText, LaTeX, Sort, Length, Max, RotateText 等命令，以及对象 f, fx, fy, l2, ax, l1, ay, fxx, fxy, fyy, fyx, Text2, M, D, Text4, f_{xx}, l4, l5, lfx, lfy, l21, text3, text1, text2, button1, l4a, l5a, l1a, Schaltfläche1, lmax, lmin, lsad, lämax, lämin, läsad, l4aa, text1a, lq, läq, z0；④适合大学数学、微积分课程教学或自学，用于理解和掌握多元函数极值点的判定方法。

GeoGebra 命令及参数

  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