Is f(x,y,z) = 8xyz for Maximizing Volume in Ellipsoid?

[mohamed el teir]

regarding question number 10, we have h = f + λg where g is the constraint (the ellipsoid) and f is the function we need to maximize or minimize (the rectangular parallelpiped volume),
now my question : is it right that f is 8xyz ? i mean if we take f to be xyz not 8xyz and solved till we got the value of xyz, the resulting value is a maximized rectangular parallelpiped in the ellipsoid in one octan only, i mean: to get the whole maximized volume we need multiply by 8, is this right ?

 

[andrewkirk]

Yes I suspect they want you to take f(x,y,z)=8xyz. Solving for that would give you the largest rectangular parallelpiped whose edges are parallel to the x, y and z axes. Solving the problem without that constraint in italics would I expect be very messy, since you'd have to allow for all possible rotations of the shape in 3D. Fortunately, I expect that the largest such shape is one that obeys that constraint anyway, although the proof of that is not immediately obvious. Since the problem would be so messy without assuming that, I expect they want you to assume it (or, more likely, they never even thought of that complication).