Question

Cho x, y, z > 0 thỏa mãn xyz = 1. CMR: x³ + y³ + z³ \(\ge\) x + y + z

Answer 1

\(x^3+1+1\ge3\sqrt[3]{x^3}=3x\); \(y^3+1+1\ge3y\); \(z^3+1+1\ge3z\)

\(\Rightarrow x^3+y^3+z^3+6\ge3\left(x+y+z\right)\ge x+y+z+2.3\sqrt[3]{xyz}=x+y+z+6\)

\(\Rightarrow x^3+y^3+z^3\ge x+y+z\)

Dấu "=" xảy ra khi \(x=y=z=1\)