Question

Chứng minh :

 \(\left(\frac{a^2}{b^2}\right)+\left(\frac{b^2}{c^2}\right)+\left(\frac{c^2}{a^2}\right)\ge\left(\frac{c}{b}\right)+\left(\frac{b}{a}\right)+\left(\frac{a}{c}\right)\)

Answer 1

Áp dụng BĐT Cô - si cho vế trái ta có

\(\Rightarrow\left\{\begin{matrix}\frac{a^2}{b^2}+\frac{b^2}{c^2}\ge\frac{2a}{c}\\\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{2b}{a}\\\frac{a^2}{b^2}+\frac{c^2}{a^2}\ge\frac{2c}{b}\end{matrix}\right.\)

Cộng theo từng vế ta có:
\(\frac{a^2}{b^2}+\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+\frac{c^2}{a^2}\ge\frac{2a}{c}+\frac{2b}{a}+\frac{2c}{b}\)

\(\Rightarrow2\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\right)\ge2\left(\frac{a}{c}+\frac{b}{a}+\frac{c}{b}\right)\)

\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{c}+\frac{b}{a}+\frac{c}{b}\) ( đpcm )