such a box has one vertex at the origin (0, 0, 0), and the vertex opposite this one is affixed to the plane with coordinates . the volume of the box is then , which we want to maximize subject to the constraint . of course, we don't want a degenerate box, so we assume each of is positive.
we can use lagrange multipliers - the lagrangian is
with partial derivatives (set equal to 0)
so the largest volume that can be attained is .
the answer is a reflection across the line m
it was 5/16 of a page long.
that would be 1 1/4 - 1 1/4 * 3/4
= 5/4 - 5/4 * 3/4
= 5/4 - 15/16
= 20/16 - 15/16
= 5/16 (answer).