函数 f(x, y) 的临界点性质：局部最大值/最小值/鞍点

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