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Finding Critical Points with Hessian Matrix

Finding Critical Points with Hessian Matrix
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2019-04-25 03:08
CalculusDerivativeMatrix
This example demonstrates how to locate and classify critical points of a multivariable function using the Hessian matrix, identifying whether they are maxima, minima, or saddle points. Key mathematical concepts include multivariable functions, partial derivatives, the Hessian matrix, and its determinant test for classification. GeoGebra commands/objects used in construction are Simplify, ImplicitCurve, Zip, Sequence, LaTeX, RotateText, and TableText. Suitable for university-level calculus and multivariable calculus courses, or self-study to deepen understanding of optimization techniques in multivariable calculus.

GeoGebra Commands & Parameters

  1. Simplify(Derivative[f, y, 1]) → fy
  2. Simplify(Derivative[fy, x, 1]) → fyx
  3. Simplify(Derivative[fy, y, 1]) → fyy
  4. Simplify(Derivative[f, x, 1]) → fx
  5. Simplify(Derivative[fx, y, 1]) → fxy
  6. Simplify(Derivative[fx, x, 1]) → fxx
  7. ImplicitCurve(fx(x, y)) → ax
  8. ImplicitCurve(fy(x, y)) → ay
  9. Zip({{fxx(a), fxy(a)}, {fyx(a), fyy(a)}}, a, l1a) → M
  10. Zip(Determinant[a], a, M) → D
  11. Zip(fxx(a), a, l1a) → f_{xx}
  12. Sequence(If[D(i) > 0, If[f_{xx}(i) > 0, "local minimum", "local maximum"], "saddle point"], i, 1, Length[l1a]) → l4
  13. Zip((x(a), y(a), f(a)), a, l1a) → l5
  14. Zip(fx(a), a, l1a) → lfx
  15. Zip(fy(a), a, l1a) → lfy
  16. LaTeX(f, true) → text2
  17. RotateText(txt1, 90°) → txt2
  18. TableText(Append["i", 1…Length[l5]], Append["critical points", l5], Append["f_{x}", lfx], Append["f_{y}", lfy], Append["f_{xx}", f_{xx}], Append["Det", D], Append["", l4], "v") → text1