Prismatic Package

From $AM \ge GM$ , we obtain:

$xy + 2yz + 2xz \ge 3 \sqrt [ 3 ]{ 4(xyz)^2 } = \sqrt [ 3 ]{ 4 \times 108^2 }$

Equality occurs when:

$xy = 2yz = 2xz \Leftrightarrow x = y = 2z$

Then:

$V = xyz = 2z \times 2z \times z = 4z^3 = 108 \Rightarrow z = 3$

Hence, $x+y+z=2z+2z+z=5z=15$

min f(x,y,z) = xy + 2yz + 2xz subject to xyz = 108.

Substitute z = $\frac{108}{xy}$ to get objective function f(x,y) = xy + $\frac{216}{x}$ + $\frac{216}{y}$

Differentiate w.r.t. x and equate to 0 to get $x^2$ y = 216.

Differentiate w.r.t. y and equate to 0 to get $y^2$ x = 216.

The above two equations yield x = y = 6 (evident from the symmetry of the objective function f that x = y, so could have reduced the minimization to a single variable problem)

Since xyz = 108, we get z = 3.

Thus, x+y+z = 15.