Question

1. Cho a, b, c >= 0.Chứng minh: \(\left(1+a\right)\left(1+b\right)\left(1+c\right)\ge\left(1+\sqrt[3]{abc}\right)^3\)

Answer 1

Ta có:

\(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge3\sqrt[3]{\frac{1}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\)

\(\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}\ge3\sqrt[3]{\frac{abc}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\)

Cộng vế với vế:

\(3\ge\frac{3\left(1+\sqrt[3]{abc}\right)}{\sqrt[3]{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\Leftrightarrow\sqrt[3]{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\ge\left(1+\sqrt[3]{abc}\right)\)

\(\Leftrightarrow\left(1+a\right)\left(1+b\right)\left(1+c\right)\ge\left(1+\sqrt[3]{abc}\right)^3\)

Dấu "=" xảy ra khi \(a=b=c\)