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

The point on the plane determines the volume of the box, since . Restricting the box to lie within the first octant is to say that .

Let's do it via Lagrange multipliers. The Lagrangian is



with partial derivatives (set equal to 0)






We have





We already assume , so we can ignore those options, leaving us with , , and . Substituting into the plane equation, we get



So the box with largest volume has its vertex (the one opposite the vertex at the origin) in the plane at , giving a volume of .

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