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Finding and Classifying Critical Points with Hessian Matrix

Finding and Classifying Critical Points with Hessian Matrix
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2019-04-25 03:20
FunctionCalculusMatrixOptimization
This example demonstrates how to use the Hessian matrix to find and classify critical points of a multivariable function, identifying whether they are local maxima, minima, or saddle points. Key mathematical concepts include multivariable functions, partial derivatives, the Hessian matrix, second partial derivatives, and their applications in optimization problems. GeoGebra commands/objects used in construction include Simplify, ImplicitCurve, Zip, Sequence, TableText, LaTeX, Sort, Length, and RotateText. Suitable for advanced calculus, multivariable function analysis courses, or self-study to understand optimization techniques for multivariable functions.

GeoGebra Commands & Parameters

  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"], "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], -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. RotateText(txt1, 90°) → txt2

AI Drawing Prompt Reference

Plot a multivariable function and find its critical points. First, compute the first-order partial derivatives and draw their level curves using implicit curves. Then, calculate the second-order partial derivatives to construct the Hessian matrix and compute its determinant to classify the critical points. Finally, categorize the critical points as local minima, maxima, or saddle points based on the sign of the determinant.
The prompt above was compiled from the original drawing commands and is for reference only.