1. ## Minima

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.

2. 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 (*) )

3. Originally Posted by Winding Function
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.
It looks to be $\displaystyle (x,y,z)=(1,1,1)$, but I'm not sure about that. I certainly couldn't prove it.