Optimization program using Lagrange multipliers.

[theBEAST]

Homework Statement

Here is the problem, the solution and my question (in red):

I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject anything right?

Thanks!

 

[voko]

I do not think your constraint really corresponds to the problem. x, y, z are dimensions of the box, not coordinates of any of its points.

 

[HallsofIvy]

Homework Statement

Here is the problem, the solution and my question (in red):

I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject anything right?

Thanks!

Yes, if the problem were just to find numbers x, y, and z, satisfying $x^2+ y^2+ z^2= r^2$, that maximize 8xyz, then negative values would also be acceptable. There would, in fact, be 8 different solutions.

Personally, I think the solution given is a little "terse". Because the original problem made no mention of x, y, and z, I would have started: "Set up a coordinate system having the center of the sphere as origin and axes parallel to the sides of the rectangular solid. Take "x", "y", and "z" to be the coordinates of the vertex in the first octant."

That would make it clear that x, y, and z are positive.