Sử dụng BĐT: \(\left(x+y+z\right)^3\ge27xyz\Rightarrow\left(\frac{x+y+z}{3}\right)^3\ge xyz\)
\(\Rightarrow\left(\frac{1+a+1+b+1+c}{3}\right)^3\ge\left(1+a\right)\left(1+b\right)\left(1+c\right)\)
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:
\(1\ge\frac{1+\sqrt[3]{abc}}{\sqrt[3]{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\Rightarrow\left(1+a\right)\left(1+b\right)\left(1+c\right)\ge\left(1+\sqrt[3]{abc}\right)^3\)
Dấu "=" 3 BĐT trên xảy ra khi \(a=b=c\)
Lại có:
\(1+\sqrt[3]{abc}\ge2\sqrt{\sqrt[3]{abc}}\Rightarrow\left(1+\sqrt[3]{abc}\right)^3\ge\left(2\sqrt{\sqrt[3]{abc}}\right)^3=8\sqrt{abc}\)Dấu "=" xảy ra khi \(a=b=c=1\)
