Solve Crazy Lagrange Problem: Min Distance from Origin

[daBish]

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Ok this is the question I had on a test today:

given this constraint equation z^2-xy+1=0 find the min. distance from the origin using Lagrange method.

so basically you use D^2=x^2+y^2+z^2 as the other equation. however, it basically goes nuts from there. especially if you set it up like you are suppose to.
Fx=(lambda)Gx
Fy=(lambda)Gy
Fz=(lambda)Gz
g=0

(capitals are partial derivatives)

with f as the distance formual and g as the constraint

this one sucks but if someone could help it would be greatly appreciated

 

[matt grime]

Minimizing D is the same as minimizing D^2, so we solve

2x = -ky
2y=-kx
2z=2kz

whence k=1, x=-y, which leads to nonsense, or z=0, and 2x=-ky=k^2x, so k=sqrt(2), also note that xy=1, and the solution follows.

edit, thanks to arildno, it should read:

2x=-ky=k^2x/2, ie

4x=xk^2, whence k=+/-2

from which you should be able to get the answer.

 

[M98Ranger]

The Lagrange method is a powerful tool for solving constrained optimization problems, but it can be quite challenging to apply, especially in more complex problems. In this case, we are trying to find the minimum distance from the origin, subject to the constraint z^2-xy+1=0. This means that we are trying to minimize the function f(x,y,z) = x^2 + y^2 + z^2, while satisfying the constraint g(x,y,z) = z^2-xy+1=0.

To solve this problem, we first set up the Lagrangian function L(x,y,z,λ) = f(x,y,z) - λg(x,y,z). This function represents the trade-off between the objective function (minimizing distance) and the constraint (satisfying the given equation). The parameter λ is known as the Lagrange multiplier, and it helps us incorporate the constraint into our optimization problem.

Next, we take the partial derivatives of L with respect to x, y, z, and λ, and set them equal to 0. This will give us a system of equations that we can solve to find the optimal values for x, y, z, and λ. Once we have these values, we can plug them back into the original objective function to find the minimum distance from the origin.

However, as you mentioned, this problem can get quite complicated and may require some advanced mathematical techniques to solve. If you are struggling with it, I would recommend seeking help from a tutor or classmate who may have a better understanding of the Lagrange method. It's always a good idea to practice solving similar problems beforehand, so you can be better prepared for the test. Good luck!