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Extrema of Functions of Two Variables

Extrema of Functions of Two Variables
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2017-04-07 21:37
FunctionCalculusOptimization
This example demonstrates the extrema of functions of two variables and their geometric significance. It involves core mathematical concepts like partial derivatives, Hessian matrix, and function extremum determination. Construction highlights include using commands like Derivative, IntersectPath, Vector, OrthogonalPlane, Length, etc. Suitable for university-level calculus and mathematics education.

GeoGebra Commands & Parameters

  1. Derivative(f, x) → f_x
  2. Derivative(f_x, y) → f_{xy}
  3. Derivative(f_x, x) → f_{xx}
  4. Derivative(f, y) → f_y
  5. Derivative(f_y, y) → f_{yy}
  6. IntersectPath(g, f_x) → ndx
  7. IntersectPath(g, f_y) → ndy
  8. Derivative(f_y, x) → f_{yx}
  9. Textfield(f) → CasillaDeEntrada1
  10. Vector(((-f_x(P_1)), (-f_y(P_1)), 1) / sqrt(1 + f_x(P_1)^(2) + f_y(P_1)^(2))) → u
  11. Vector(P_1, P_1 + u) → u'
  12. OrthogonalPlane(P_1, u') → a
  13. IntersectPath(a, f) → b
  14. Textfield(P) → CasillaDeEntrada2
  15. Length(lPc) → n
  16. Join({{{"(x, y)", "f(x,y)", "f_x(x,y)", "f_y(x,y)", "f_{xx}(x,y)", "H(x,y)", "Tipo"}}, Sequence[{lPc(k), f(lPc(k)), f_x(lPc(k)), f_y(lPc(k)), f_{xx}(lPc(k)), H(lPc(k)), If[f_x(lPc(k)) ≠ 0 ∨ f_y(lPc(k)) ≠ 0, "No crítico", f_{xx}(lPc(k)) > 0 ∧ H(lPc(k)) > 0, "Mínimo relativo", f_{xx}(lPc(k)) < 0 ∧ H(lPc(k)) > 0, "Máximo relativo", H(lPc(k)) ≠ 0, "Punto de silla", "??"]}, k, 1, Length[lPc]]}) → tPc
  17. TableText(tPc, "|_") → texto2
  18. Sequence((x(lPc(k)), y(lPc(k)), f(lPc(k))), k, 1, n) → lPcz
  19. Sequence({lPc(k), f_x(lPc(k)), f_y(lPc(k)), f_{xx}(lPc(k)), H(lPc(k)), If[f_x(lPc(k)) ≠ 0 ∨ f_y(lPc(k)) ≠ 0, "No crítico", f_{xx}(lPc(k)) > 0 ∧ H(lPc(k)) > 0, "Mínimo relativo", f_{xx}(lPc(k)) < 0 ∧ H(lPc(k)) > 0, "Máximo relativo", H(lPc(k)) ≠ 0, "Punto de silla", "??"]}, k, 1, Length[lPc]) → lPc0