G use lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordin
ate planes and one vertex in the given plane.x + 9y + 8z = 27
1 answer:
This basically comes down to maximizing

subject to

and enforcing

. We have the Lagrangian

with partial derivatives (set equal to 0 to find critical points)

Solving the first equation for

gives

. Substituting this into the next two equations, we have


Now



So the vertex of the cuboid in the given plane that maximizes the cuboids volume is

, giving a volume of

.
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