Question

Câu 1: cho a,b,c là các số thực dương. Chứng minh rằng: (\(1+\dfrac{a}{b})\)\((1+\dfrac{b}{c})\)\((1+\dfrac{c}{a})\) ≥ 8

Answer 1

ta có : \(\left(1+\dfrac{a}{b}\right)\left(1+\dfrac{b}{c}\right)\left(1+\dfrac{c}{a}\right)=\left(1+\dfrac{b}{c}+\dfrac{a}{b}+\dfrac{ab}{bc}\right)\left(1+\dfrac{c}{a}\right)\)

\(=1+\dfrac{c}{a}+\dfrac{b}{c}+\dfrac{bc}{ac}+\dfrac{a}{b}+\dfrac{ac}{ba}+\dfrac{ab}{bc}+1\)

\(=2+\left(\dfrac{c}{a}+\dfrac{ab}{bc}\right)+\left(\dfrac{b}{c}+\dfrac{ac}{ba}\right)+\left(\dfrac{a}{b}+\dfrac{bc}{ac}\right)\ge2+2+2+2=8\) \(\Rightarrowđpcm\)