- #1

The Head

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- Homework Statement
- Consider a particle that moves in x^2+2y^2+2z^2<=4 and is subject to the force field f(x,y,z)= (1,-2z,-2y). If f=grad(F), find the values in this region that maximize f.

- Relevant Equations
- grad(F)=0 to find extrema

Lagrange Mulitpliers Technique

I found that f= x -2yz. To maximize f, I can first inspect the solutions to grad(F)=0. z=y=0 pops out, but I'm not sure what to do with the x-component equaling 1. Do we just include (x,0,0) as a solution? I think the problem wants specifics though, based on what I've seen previously from problem sets by this instructor.

Using Lagrange Multipliers, we get:

1=2xλ

-2z=4yλ

-2y=4zλ

Working through this, I find that z(1-4λ)=0. If z=0, then y=0. If λ=1/4, then y=-z/2 and x=2. So it appears we have (x,0,0) (as before) and (2, -z/2, z).

I'm not sure how to work through these to get the specifics here. Am I missing something, or is this a departure from this instructor's typical problems? Also, I don't think I need to parameterize and look for points inside the boundary here because of the Laplacian being equal to zero (I read that somewhere, but could be mistaken).

Using Lagrange Multipliers, we get:

1=2xλ

-2z=4yλ

-2y=4zλ

Working through this, I find that z(1-4λ)=0. If z=0, then y=0. If λ=1/4, then y=-z/2 and x=2. So it appears we have (x,0,0) (as before) and (2, -z/2, z).

I'm not sure how to work through these to get the specifics here. Am I missing something, or is this a departure from this instructor's typical problems? Also, I don't think I need to parameterize and look for points inside the boundary here because of the Laplacian being equal to zero (I read that somewhere, but could be mistaken).