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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likvau · Accepted Answer

Definition used:  &quot; The Lagrange multipliers defined as ∇f(x,y,z)=λ∇g(x,y,z). This equation can be expressed as fx=λgx,fy=λgy,fz=λgzandg(x,y,z)=k&quot;  Calculation:  Let the volume of the rectangular box be V=f(x,y,z)=xyz where x&amp;gt;0,y&amp;gt;0,z&amp;gt;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)=λ∇g(x,y,z) is computed as follows,  ∇f(x,y,z)=λ∇g(x,y,z)  ⟨fx,fy,fz⟩=λ⟨gx,gy,gz⟩  ⟨fx(xyz)fy(xyz)fz(xyz)⟩=λ⟨gx(x+2y+3z)gy(x+2y+3z)gz(x+2y+3z)⟩  ⟨yz,xz,xy⟩=λ⟨1,2,3⟩  Thus, the value of ∇f(x,y,z)=λ∇g(x,y,z)is⟨yz,xz,xy⟩=λ⟨1,2,3⟩.  By the definition, ⟨yz,xz,xy⟩=λ⟨1,2,3⟩ 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

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.

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