Then xyz=8, and find the minimum value of f(x, y, z)=xy+2yz+2xz.

Let L(x, y, z)=xy+2yz+2xz+a(xyz-8) by Lagrange multiplier method.

Lx=y+2z+ayz=0

Ly=x+2z+axz=0

Lz=2y+2x+axy=0

La=xyz-8=0

From the above formula, we can get the two-thirds power of x=y=4 and the negative one-third power of z=2x4, which is what we want.