Question

a,b,c khác 0. Chứng minh:

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

Answer 1

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

Cộng vế với vế

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

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

Dấu "=" khi \(a=b=c\)