I am trying to solve this problem but I do not know how to even start it.
Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $x + 2y + 3z = 9$.
Can anyone help me? Any help I would appreciate a lot! thanks!
The hint.
By AM-GM $$9=x+2y+3z\geq3\sqrt[3]{x\cdot2y\cdot3z}=3\sqrt[3]{6xyz}.$$ The equality occurs for $x=2y=3z$ and $x+2y+3z=9.$
I got $4.5$.