The volume of the largest rectangular box which lies in the first octant with three faces in the coordinate planes and its one of the vertex in the plane displaystyle{x}+{2}{y}+{3}{z}={6} by using Lagrange multipliers.

Jaden Easton 2021-01-28 Answered
The volume of the largest rectangular box which lies in the first octant with three faces in the coordinate planes and its one of the vertex in the plane x+2y+3z=6 by using Lagrange multipliers.
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Expert Answer

likvau
Answered 2021-01-29 Author has 75 answers
Definition used:
" The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z). This equation can be expressed as fx=λgx,fy=λgy,fz=λgzandg(x,y,z)=k"
Calculation:
Let the volume of the rectangular box be V=f(x,y,z)=xyz where x>0,y>0,z>0.
Thus, the maximize function f(x,y,z)=xyz subject to the constraint g(x,y,z)=x+2y+3z=6.
The Lagrange multipliers f(x,y,z)=λg(x,y,z) is computed as follows,
f(x,y,z)=λg(x,y,z)
fx,fy,fz=λgx,gy,gz
fx(xyz)fy(xyz)fz(xyz)=λgx(x+2y+3z)gy(x+2y+3z)gz(x+2y+3z)
yz,xz,xy=λ1,2,3
Thus, the value of f(x,y,z)=λg(x,y,z)isyz,xz,xy=λ1,2,3.
By the definition, yz,xz,xy=λ1,2,3 can be expressed as follows,
yz=λ(1)
xz=2λ(2)
xy=3λ(3)
From the equations (1), (2) and (3),
yz=xz2=xy3
Consider yz=xz2 and compute the value of y,
yz=xz2

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