Question

1/a +1/b+1/c >+ 1/√ab + 1/√bc +1√ca

Answer 1

\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge?\dfrac{1}{\sqrt{ab}}+\dfrac{1}{\sqrt{bc}}+\dfrac{1}{\sqrt{ca}}\\ \Leftrightarrow\dfrac{2}{a}+\dfrac{2}{b}+\dfrac{2}{c}\ge\dfrac{2}{\sqrt{ab}}+\dfrac{2}{\sqrt{bc}}+\dfrac{2}{\sqrt{ca}}\)

Áp dụng bất đẳng thức AM-GM ta có :

\(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{2}{\sqrt{ab}}\\ \dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{2}{\sqrt{bc}}\\ \dfrac{1}{c}+\dfrac{1}{a}\ge\dfrac{1}{\sqrt{ca}}\)

Vậy \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{1}{\sqrt{ab}}+\dfrac{1}{\sqrt{bc}}+\dfrac{1}{\sqrt{ca}}\left(đpcm\right)\)