giả sử \(a+\frac{1}{a}\ge2\)
vì a > 0 => \(a^2+1\ge2a\)
<=> \(a^2+1-2a\ge0\)
<=> \(\left(a-1\right)^2\ge0\)( luôn đúng vs mọi a > 0)
=> \(a+\frac{1}{a}\ge2\). CMTT ta có \(b+\frac{1}{b}\ge2\)và \(c+\frac{1}{c}\ge2\)(1)
Ta có \(\left(a+1\right)\left(b+1\right)\left(c+1\right)=abc+ac+bc+ab+a+b+c+1\)
\(=1+1+\frac{1}{b}+\frac{1}{a}+\frac{1}{c}+a+b+c\)\(=2+\left(\frac{1}{a}+a\right)+\left(\frac{1}{b}+b\right)+\left(\frac{1}{c}+c\right)\)
Từ (1) =>\(2+\left(\frac{1}{a}+a\right)+\left(\frac{1}{b}+b\right)+\left(\frac{1}{c}+c\right)\ge8\)(đpcm)