The problem is to optimize (find the maximum The problem is to maximiz the function
f(x,y,z)=xyz
With the constrain
2(xy + xz + yz)=64; xy+xz+yz=32
Using the Lagrange Multipliers
F(x,y,z) = xyz - £(xy+xz +yz-32)
Deriving with respect to x:
yz - £(y+z)=0 ....i
Deriving with respect to y:
xz - £(x+z)=0 ...ii
Deriving with respect to z:
xy - 2£(x+y)=0 ....iii
Deriving with respect to £:
xy+xz+yz=32 .....iv
From (i) and (ii)
yz/2(y+z) = xz/2(x+z)
y/(y+z) = x/(x+z)
yx+yz=xy+xz
y=x
From (i) and (iii)
x=z
So, from (iv)
x^2+x^2+x^2=32
x^2=32/3
x=y=z=sqrt (32/3)
Vmax = sqrt (32/3)^3