If the sides of the box in the directions of the three coordinate axes have lengths x, y and z, then the volume of the box is xyz. The only vertex of the box that does not lie in one of the coordinate planes is the one at (x,y,z), and that has to satisfy 2x + 3y + 4z = 6. So you should use the method of Lagrange multipliers to maximise xyz subject to the constraint 2x + 3y + 4z = 6.