Question: Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoidx29+y281+z264=1 By symmetry, you can

Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoidx29+y281+z264=1 By symmetry, you can restrict your attention to the first octant (where x,y,z0), and assume your volume has the form V=8xyzThen arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume:

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