Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid $$\frac{x^2}{4} + \frac{y^2}{36} + \frac{z^2}{64} = 1$$ Hint: By symmetry, you can restrict your attention to the first octant (where $x, y, z \ge 0$), and assume your volume has the form $V = 8xyz$. Then arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume: