2. Let $E =\{(x,y,z) ∈\mathbb{ R}^{3} | x,y,z > 0,xy + yz + zx = 1\}$

Question

2. Let $E = \{(x,y,z) \in \mathbb{R}^{3} | x,y,z > 0,xy + yz + zx = 1\}$. Prove that there exists $(a,b,c) \in E$ such that $abc ≥ xyz$, for all $(x,y,z) \in E$.

Answer 1

Hint:  by AM-GM $\;\sqrt[3]{x^2y^2z^2} \le \dfrac{xy+yz+zx}{3}=\dfrac{1}{3}\;$ so $\;xyz \le \sqrt{\dfrac{1}{3^3}}\,$, with equality iff $\,x=y=z\,$.