Find the minimal value of the product $\displaystyle P(x, y, z) = (x+y)(y+z)(z+x)$, when $\displaystyle xyz=1$ and $\displaystyle x, y, z $ are positive real numbers.
We have: $\displaystyle
x + y \ge 2 \cdot \sqrt {xy}
$ ( just see that $\displaystyle
\left( {\sqrt x - \sqrt y } \right)^2 \ge 0
$ (*) and expand)
Similarly with the others. And multiply: $\displaystyle
\left( {x + y} \right) \cdot \left( {x + z} \right) \cdot \left( {y + z} \right) \ge 2^3 \cdot \sqrt {xy} \sqrt {xz} \sqrt {yz}
$
And note that: $\displaystyle
\sqrt {xy} \sqrt {xz} \sqrt {yz} = xyz = 1
$ thus: $\displaystyle
\left( {x + y} \right) \cdot \left( {x + z} \right) \cdot \left( {y + z} \right) \ge 8
$
Equality is achieved iff: $\displaystyle
x = y = z =1
$ ( see (*) )