MathLabs

Problem 3

Let x,y,zx,y,z be positive real numbers and let w=xyz3w=\sqrt[3]{xyz}. Prove that (1+x/y)(1+y/z)(1+z/x)≥2+2(x+y+z)/w(1+x/y)(1+y/z)(1+z/x)\ge2+2(x+y+z)/w.
Step 3 of 4: Apply AM-GM to x,y,zx,y,z
Previous stepNext step
(x+y+z)/w≥3(x+y+z)/w\ge3
Detailed analysis

A second AM-GM inequality gives x+y+z≥3xyz3=3wx+y+z\ge3\sqrt[3]{xyz}=3w, hence (x+y+z)/w≥3(x+y+z)/w\ge3.