We'll assume that the box is suppose to lie in the first octant. This seems like a reasonable starting point.

If the vertex in the plane has coordinates then the volume of the box is .

So the problem boils down to maximising the quantity subject to the conditions that , , are positive and .

Now try the AM-GM inequality, namely with equality iff .

You should now be able to deduce that the maximum volume is 144. Happy hunting.