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Finding Critical Points and Their Nature Using Hessian Matrix

Finding Critical Points and Their Nature Using Hessian Matrix
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2019-04-25 02:59
FunctionCalculusDerivative
This example demonstrates how to use the Hessian matrix to find the critical points and their nature of a function. Key mathematical concepts include derivatives, Hessian matrix, and function extrema. Using commands like Simplify, Zip, and Sequence, the example constructs the second derivative matrix of the function and determines the nature of the critical points. Suitable for calculus and advanced mathematics courses.

GeoGebra Commands & Parameters

  1. Simplify(Derivative[f, x, 1]) → fx
  2. Simplify(Derivative[fx, x, 1]) → fxx
  3. Simplify(Derivative[fx, y, 1]) → fxy
  4. Simplify(Derivative[f, y, 1]) → fy
  5. Simplify(Derivative[fy, y, 1]) → fyy
  6. Simplify(Derivative[fy, x, 1]) → fyx
  7. Zip({{fxx(a), fxy(a)}, {fyx(a), fyy(a)}}, a, l1) → M
  8. Zip(Determinant[a], a, M) → D
  9. Zip(fxx(a), a, l1) → f_{xx}
  10. Sequence(If[D(i) > 0, If[f_{xx}(i) > 0, "local minimum", "local maximum"], "saddle point"], i, 1, Length[l1]) → l4
  11. Zip((x(a), y(a), f(a)), a, l1) → l5
  12. Zip(fx(a), a, l1) → lfx
  13. Zip(fy(a), a, l1) → lfy
  14. 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
  15. LaTeX(f, true) → text3
  16. RotateText(txt1, 90°) → txt2

AI Drawing Prompt Reference

Plot the function f(x, y) and find its critical points. Use the Hessian matrix to determine the nature of these points, including the second-order partial derivatives matrix, determinant, local minimum, local maximum, and saddle point.
The prompt above was compiled from the original drawing commands and is for reference only.