MathLabs

Problem 4

Let x,y,zx,y,z be positive numbers such that 1x+1y+1z=1\frac1x+\frac1y+\frac1z=1. Show that x+yz+y+zx+z+xy≥xyz+x+y+z\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}.
Step 8 of 8: Conclude
Previous stepNext step
L2−R2≥0  ⟹  L≥R.L^2-R^2\ge0\implies L\ge R.
Detailed analysis

The non-cross terms in L squared and R squared agree because xy+yz+zx=xyz. The summed estimate handles all cross terms, so L squared is at least R squared. Equality holds at x=y=z=3.