Question

Cho các số dương a,b,c CMR

\(\frac{7}{a}+\frac{5}{b}+\frac{4}{c}\ge4\left(\frac{4}{a+b}+\frac{1}{b+c}+\frac{3}{c+a}\right)\)

Answer 1

Áp dụng bđt \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\left(x;y>0\right)\) (tự c/m ha)

\(\frac{7}{a}+\frac{5}{b}+\frac{4}{c}=\left(\frac{4}{a}+\frac{4}{b}\right)+\left(\frac{1}{b}+\frac{1}{c}\right)+\left(\frac{3}{a}+\frac{3}{c}\right)\)

                               \(=4\left(\frac{1}{a}+\frac{1}{b}\right)+\left(\frac{1}{b}+\frac{1}{c}\right)+3\left(\frac{1}{a}+\frac{1}{c}\right)\)

                               \(\ge4.\frac{4}{a+b}+\frac{4}{b+c}+3.\frac{4}{a+c}=4\left(\frac{4}{a+b}+\frac{1}{b+c}+\frac{3}{c+a}\right)\)

Dấu "=" <=> a = b = c