₂ΛM 1: Lagrange Multipliers with One Constraint. Examples
1. Lagrange Multipliers ● Example 1
2. Lagrange Multipliers ● Example 2
3. Lagrange Multipliers ● Example 3
4. Lagrange Multipliers ● Example 4
5. Lagrange Multipliers ● Example 5
6. Lagrange Multipliers ● Example 6
7. Lagrange Multipliers ● Example 7
8. Lagrange Multipliers ● Example 8
9. Directional derivatives and Gradient
10. Lagrange Points 1
11. LagrangeOptimization-2
12. Gradient and Directional Derivatives
13. Extremwertaufgabe f(x,y)
14. The Gradient Vector and Directional Derivative
15. Vector Fields and Line Integrals
16. 勾配ベクトル場の性質（2014）
17. Gradient vector field
18. GeoGebra Tutorial - Lagrange Multiplier

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₂ΛM 1: Lagrange Multipliers with One Constraint. Examples

Roman Chijner, Jul 12, 2020

[math][/math] How to find the maxima and minima of the function f (x, y) under the constraints g (x, y) = c? A necessary condition for such a “critical point” is that the gradients of f and g are parallel. The system of equations: ∇f (x, y) = λ∇g (x, y), g (x, y) = c with three unknowns x, y, λ are called the Lagrange equations. The variable λ is called the Lagrange multiplier. The equations are represented as two implicit functions. Points of intersections are solutions.They are provided using CAS and GGB commands. All results are shown in the tables. The results can be graphically compare and choose the best. The cases λ = 0 correspond to critical points only for the function f (x, y). Searching and exploring critical points in the examples you will find in "Critical points and their nature using hessian matrix"- [url]https://www.geogebra.org/m/krmmnhu4[/url] *From Book: Extended definitions of point location estimates [url]https://www.geogebra.org/m/hhmfbvde[/url] From: List of My Public Books on GeoGebra Topics: Constructing polyhedra -[url]https://www.geogebra.org/m/eabstecp[/url]

1. Lagrange Multipliers ● Example 1
2. Lagrange Multipliers ● Example 2
3. Lagrange Multipliers ● Example 3
4. Lagrange Multipliers ● Example 4
5. Lagrange Multipliers ● Example 5
6. Lagrange Multipliers ● Example 6
7. Lagrange Multipliers ● Example 7
8. Lagrange Multipliers ● Example 8
9. Directional derivatives and Gradient
10. Lagrange Points 1
11. LagrangeOptimization-2
12. Gradient and Directional Derivatives
13. Extremwertaufgabe f(x,y)
14. The Gradient Vector and Directional Derivative
15. Vector Fields and Line Integrals
16. 勾配ベクトル場の性質（2014）
17. Gradient vector field
18. GeoGebra Tutorial - Lagrange Multiplier

1.

Lagrange Multipliers ● Example 1

[size=85] Extremize a function of two variables f(x,y) =2x[sup]2[/sup]+2y[sup]2[/sup]+8 under a constraint qiven by equality g: x[sup]2[/sup]+2y[sup]2[/sup]-6x=7. CAS and ggB solutions are compared. The critical points of the corresponding Lagrangian, its global maximum and minimum are determined.[/size]

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₂ΛM 1: Lagrange Multipliers with One Constraint. Examples

Lagrange Multipliers ● Example 1

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