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Critical Points as Geometric Medians and Hessian Matrix

Critical Points as Geometric Medians and Hessian Matrix
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2020-12-14 23:28
FunctionCalculusOptimization
In collections ₅Point Location estimators: Geometric Medians on a sphere
This example explores the nature of critical points using the concept of geometric medians and the Hessian matrix. It first constructs a series of geometric objects, such as points and line segments, to demonstrate how to find the critical points of a function. Then, it uses the Hessian matrix to determine the nature of these critical points, such as whether they are maxima, minima, or saddle points. The example utilizes commands like Polygon, PointIn, Sequence, Zip, and TableText in GeoGebra, making it suitable for high school mathematics teachers and students studying calculus and optimization problems.

GeoGebra Commands & Parameters

  1. Polygon({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → q
  2. PointIn(q) → P9
  3. PointIn(q) → A_2
  4. 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
  5. Zip((x(a), y(a)), a, l1) → l5
  6. 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
  7. Function((q_f * f(u, v)) + b_0, u, (-π), π, v, ((-π)) / 2, π / 2) → f0
  8. Length(lsad) → läsad
  9. Length(lmax) → lämax
  10. Length(lmin) → lämin
  11. Zip(Segment[A, p], p, lP) → seg
  12. Sum(seg) → sum_seg
  13. Flatten(Sequence[Sequence[((-π) + ((2 * π / na) * i), ((-π)) / 2 + (π / na * j)), j, 1, na - 1], i, 1, na - 1]) → ltest_2
  14. Zip((R; x(a); y(a)), a, ltest_2) → ltest_3
  15. TableText(Append["i", Sequence[Length[lmax]]], Append["P_{Max }\;(φ,θ)", lmax], Append["(f'x,f'y)", lZf'xymax], "v") → textmax
  16. TableText(Append["i", Sequence[Length[(lmin * x)]]], Append["P_{Max }\;(φ,θ)", lmin], Append["(f'x,f'y)", lZf'xymin], "v") → textmin
  17. TableText(Append["i", Sequence[Length[(lsad * x)]]], Append["P_{Max }\;(φ,θ)", lsad], Append["(f'x,f'y)", lZf'xysad], "v") → textsad
  18. Textfield(R) → InputBox1
  19. Textfield(n) → InputBox2
  20. Textfield(lP) → InputBox3

AI Drawing Prompt Reference

Draw a rectangular region with x from -π to π and y from -π/2 to π/2. Place two random points inside. Generate a grid of test points. Display a function surface over the region. Mark critical points and list them in a table with coordinates, first and second partial derivatives, and determinant, classifying each as local maximum, minimum, or saddle. Connect each critical point to a reference point with line segments and show their total length. Include three input fields for adjusting radius, number of points, and reference point set.
The prompt above was compiled from the original drawing commands and is for reference only.