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