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Critical Points as Geometric Medians and Hessian Matrix Analysis

Critical Points as Geometric Medians and Hessian Matrix Analysis
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2019-05-30 19:13
FunctionCalculusDerivativeOptimizationGeometric Construction
This example demonstrates how to visualize and analyze critical points of a function as geometric medians using GeoGebra, focusing on the nature of these points through the Hessian matrix. Key math concepts include critical points, Hessian matrix, geometric median, and function optimization. GeoGebra commands like Polygon, PointIn, First, Sum, Sequence, Zip, TableText, Function, Length, Flatten, and Textfield are utilized for construction. Suitable for university-level mathematics students or those interested in function optimization and geometric properties.

GeoGebra Commands & Parameters

  1. Polygon({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → q
  2. PointIn(q) → aP1
  3. PointIn(q) → aP2
  4. PointIn(q) → aP3
  5. PointIn(q) → aP4
  6. PointIn(q) → aP5
  7. PointIn(q) → aP6
  8. PointIn(q) → aP7
  9. PointIn(q) → aP8
  10. PointIn(q) → P9
  11. PointIn(q) → aP10
  12. First({P1, P2, P3, P4, P5, P6, P7, P8, P9, P10}, n) → lP
  13. PointIn(q) → A_2
  14. Sum(Zip[sqrt((x(a) - ((R * cos(x)) * cos(y)))^(2) + (y(a) - ((R * sin(x)) * cos(y)))^(2) + (z(a) - (R * sin(y)))^(2)), a, lP]) → f
  15. 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
  16. Zip((x(a), y(a)), a, l1) → l5
  17. 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
  18. Function((q_f * f(u, v)) + b_0, u, (-π), π, v, ((-π)) / 2, π / 2) → f0
  19. Length(lsad) → läsad
  20. Length(lmax) → lämax
  21. Length(lmin) → lämin
  22. Zip(Segment[A, p], p, lP) → seg
  23. Sum(seg) → sum_seg
  24. Flatten(Sequence[Sequence[((-π) + ((2 * π / na) * i), ((-π)) / 2 + (π / na * j)), j, 1, na - 1], i, 1, na - 1]) → ltest_2
  25. Zip((R; x(a); y(a)), a, ltest_2) → ltest_3
  26. TableText(Append["i", Sequence[Length[lmax]]], Append["P_{Max }\;(φ,θ)", lmax], Append["(f'x,f'y)", lZf'xymax], "v") → textmax
  27. TableText(Append["i", Sequence[Length[(lmin * x)]]], Append["P_{Max }\;(φ,θ)", lmin], Append["(f'x,f'y)", lZf'xymin], "v") → textmin
  28. TableText(Append["i", Sequence[Length[(lsad * x)]]], Append["P_{Max }\;(φ,θ)", lsad], Append["(f'x,f'y)", lZf'xysad], "v") → textsad
  29. Textfield(R) → InputBox1
  30. Textfield(n) → InputBox2
  31. Textfield(lP0) → InputBox3