How to use Lagrange multipliers to find the maximum and minimum values?

Question

How do I use Lagrange multipliers to find the maximum and the minimum values of the function F(x,y) = x²y²z²

subject to the constraint function x² + y² + z² = 1?

Answer 1

optimize f(x,y,z) = x²y²z²

over the domain g(x,y,z) = x² + y² + z² - 1 = 0



∇f = λ ∇g

2xy²z² = λ 2x

2yx²z² = λ 2y

2zx²y² = λ 2z

⇒ x² = y² = z² ..... is a stationary point

3x² = 1 ........ plug into g(x,y,z)

⇒ x = y = z = ± 1/√3 ..... max

⇒ x = 0, or y = 0, or z = 0 ......... min



Answer: f_max = 1/27 at (± 1/√3, ± 1/√3, ± 1/√3) ; f_min = 0 at (0,±√(1-z²),z), (x,0,±√(1-x²)), (x,±√(1-x²),0)

Answer 2

The goal is to find where the normal of the two surfaces are parallel, that will be the maximum and minimum (maybe saddle as well) if I remember my analysis course.



Using lagrange multipliers this is expressed as a number of equations of which has to be fulfilled to reach the goal above.



The equations consist of the partial derivates with respect to x,y,z of F(x,y,z) and the constraint G(x,y,z)=x^2 + y^2 + z^2 -1



To simplify F_x means F differentiated with respect to x, and m is the lagrange multiplier. With that notation you get these equations:



F_x=m*G_x

F_y=m*G_y

F_z=m*G_z



You also have the constraint:

x^2 + y^2 + z^2=1, which also must be fulfilled.



So in total you have 4 equations from which you can find a number of points and the lagrangian multiplier.



In your case you get:



2x*y^2*z^2=2m*x

2y*x^2*z^2=2m*y

2z*y^2*x^2=2m*z

x^2 + y^2 + z^2=1



The three first implies that: y^2=x^2=z^2 => y=x=z



Inserting that for each variable in the last equation



you get:

3x^2=1=>x=(+-)*(1/3)^(1/2)



By using y=x=z you get two points

-((1/3)^(1/2), (1/3)^(1/2), (1/3)^(1/2))



and



((1/3)^(1/2), (1/3)^(1/2), (1/3)^(1/2))



I'm a little unsure of how to check if they are min, max or saddle. For two variables it's pretty straightforward. Maybe you could develop the taylor series around the points and see what happens if you increase/decrease the values of the points slightly.

Answer 3

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RE:

How to use Lagrange multipliers to find the maximum and minimum values?

How do I use Lagrange multipliers to find the maximum and the minimum values of the function F(x,y) = x²y²z²



subject to the constraint function x² + y² + z² = 1?

Answer 4

∇F(x,y,z) = λ·∇G(x,y,z) Where F(x,y,z) is the function you want to maximize and G(x,y,z) = C, where C is a constant, is the constraint on the domain. ∇F(x,y,z) = <2xy²z², 2x²yz², 2x²y²z> ∇G(x,y,z) = <2x, 2y, 2z> So we know that <2xy²z², 2x²yz², 2x²y²z> = λ·<2x, 2y, 2z> Or, written in simultaneous equations, 2xy²z² = λ·2x 2x²yz² = λ·2y 2x²y²z = λ·2z along with the constraint G(x,y,z) x² + y² + z² = 1 Solve for self-consistent sets (x,y,z,λ) that satisfy all four equations... ===== Solve ======= 2xy²z² = λ·2x 2x²yz² = λ·2y 2x²y²z = λ·2z x² + y² + z² = 1 xy²z² = λx x²yz² = λy x²y²z = λz x² + y² + z² = 1