Áp dụng bđt Cauchy:
\(ab+\frac{a}{b}\ge2a\)
\(ab+\frac{b}{a}\ge2b\)
\(\frac{a}{b}+\frac{b}{a}\ge2\)
Cộng theo vế: \(2\left(ab+\frac{a}{b}+\frac{b}{a}\right)\ge2\left(a+b+1\right)\Leftrightarrow ab+\frac{a}{b}+\frac{b}{a}\ge a+b+1\)
Dấu "=" xảy ra khi \(a=b=1\)