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Lagrange Multipliers for Conditional Extrema

Lagrange Multipliers for Conditional Extrema
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2017-05-25 16:49
VectorCoordinate GeometryCalculusDerivativeOptimization
This example demonstrates the solution of conditional extrema using the Lagrange multiplier method. It visually illustrates the principle of the multiplier method through the graphical representation of the objective function and constraint. Key mathematical concepts include conditional extrema, Lagrange multipliers, gradient vectors, and the geometric meaning of functions. Construction highlights the use of GeoGebra commands and objects such as Derivative (derivative), Point (point), Vector (vector), SelectedElement (selected element), CurveCartesian (Cartesian curve), Segment (segment), Tangent (tangent), and others like g3, g2, ell3, ell2, ell1, g1, listg, zyl_1, f, w_1, w_2, w_3, w_5, f_x, P_1, f_y, g_x, g_y, gradf_{P1}, Text1, w_4, g, b, P_2, P_3, gradf_{P2}, gradg_{P2}, gradg_{P3}, gradf_{P3}, zyl_2, zyl_3, gradg_{P1}, Text3, k21, k22, S_1, s_1, S_2, S_3, s_2. Suitable for high school and above students learning optimization problems.

GeoGebra Commands & Parameters

  1. Derivative(f, x) → f_x
  2. Point(ell1) → P_1
  3. Derivative(f, y) → f_y
  4. Derivative(g1, x) → g_x
  5. Derivative(g1, y) → g_y
  6. Vector((f_x(x(P_1), y(P_1)), f_y(x(P_1), y(P_1)))) → gradf_{P1}
  7. SelectedElement(listg) → g
  8. SelectedIndex(listg) → b
  9. Point(ell2) → P_2
  10. Point(ell3) → P_3
  11. Vector((f_x(x(P_2), y(P_2)), f_y(x(P_2), y(P_2)))) → gradf_{P2}
  12. Vector((g_x(x(P_2), y(P_2)), g_y(x(P_2), y(P_2)))) → gradg_{P2}
  13. Vector((g_x(x(P_3), y(P_3)), g_y(x(P_3), y(P_3)))) → gradg_{P3}
  14. Vector((f_x(x(P_3), y(P_3)), f_y(x(P_3), y(P_3)))) → gradf_{P3}
  15. Vector((g_x(x(P_1), y(P_1)), g_y(x(P_1), y(P_1)))) → gradg_{P1}
  16. CurveCartesian(t, sqrt(1 - t^(2) / 2), 1 / 2 - (3 / 4 * t^(2)), t, -2, 2) → k21
  17. CurveCartesian(t, (-sqrt(1 - t^(2) / 2)), 1 / 2 - (3 / 4 * t^(2)), t, -20, 2) → k22
  18. Segment(P_1, S_1) → s_1
  19. Segment(P_2, S_2) → s_2
  20. Segment(P_3, S_3) → s_3
  21. CurveCartesian(cos(t), sin(t), f(cos(t), sin(t)), t, 0, (2 * pi)) → k1
  22. CurveCartesian((sqrt(2) * cos(t)), sin(t), f((sqrt(2) * cos(t)), sin(t)), t, 0, (2 * pi)) → k2
  23. CurveCartesian(cos(t), (sqrt(2) * sin(t)), f(cos(t), (sqrt(2) * sin(t))), t, 0, (2 * pi)) → k3
  24. Tangent(S_1, k1) → t_1
  25. Segment(S_1 - UnitVector[t_1], S_1 + UnitVector[t_1]) → t11
  26. Tangent(S_2, k2) → t_2
  27. Segment(S_2 - UnitVector[t_2], S_2 + UnitVector[t_2]) → t21
  28. Tangent(S_3, k3) → t_3
  29. Segment(S_3 - UnitVector[t_3], S_3 + UnitVector[t_3]) → t31