3. Find the Volume of the largest rectangular box in the first octant with three faces in the coordinates and one vertex in the plane 2x+y + 3z = 6.

V(x,y,z) = xyz; 2x+y+3z = 6

The gradiant of V(x,y,z) = <yz, xz, xy>

The gradient of the constraint = <2, 1, 3>

2λ = yz

λ = xz

3λ = xy

x^2 + y^2 + z^2 = 12

Given the second equation the first and third equations becmoe

2xz = yz and 3xz = xy respectively.

Solving the second equation you get

2xz - yz = 0

z(2x-y) = 0

z = 0 2x - y = 0

y = 2x

plugging this into the constraint you get:

2x + 2x = 6

4x = 6

x = 3/2

this gets you y = 3 so the first point is (3/2,3,0)

Solving the third equation you get

3xz - xy = 0

x(3z-y) = 0

x = 0 3z - y = 0

y = 3z

plugging this into the constraint you get:

3z+3z = 6

6z = 6

z = 1

this gets you y = 3 so you get a point of (0, 3, 1)

V(3/2,3,0) = 0

v(0,3,1) = 0

what did I do wrong?