Giải:
Ta có: \(A=\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}=\left(1-\frac{1}{a+1}\right)+\left(1-\frac{1}{b+1}\right)+\left(1-\frac{1}{c+1}\right)\)
\(\Leftrightarrow A=3-\left(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\right)\)
Áp dụng BĐT Cauchy-Schwarz:
\(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{9}{a+b+c+3}=\frac{9}{4}\)
Suy ra \(A\leq 3-\frac{9}{4}=\frac{3}{4}\) (đpcm)
Dấu bằng xảy ra khi \(a=b=c=\frac{1}{3}\)
Xét: \(\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\)
\(\Leftrightarrow\frac{a}{2a+b+c}+\frac{b}{a+2b+c}+\frac{c}{a+b+2c}\)
Áp dụng bất đẳng thức \(\frac{1}{a+b}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}\right)\) với a , b > 0
\(\Rightarrow\left\{\begin{matrix}\frac{a}{2a+b+c}=\frac{a}{a+b+a+c}\le\frac{a}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)\\\frac{b}{a+2b+c}=\frac{b}{a+b+b+c}\le\frac{b}{4}\left(\frac{1}{a+b}+\frac{1}{b+c}\right)\\\frac{c}{a+b+2c}=\frac{c}{a+c+b+c}\le\frac{c}{4}\left(\frac{1}{a+c}+\frac{1}{b+c}\right)\end{matrix}\right.\)
\(\Rightarrow VT\le\frac{a}{4}\left(\frac{1}{a+b}+\frac{1}{a+c}\right)+\frac{b}{4}\left(\frac{1}{a+b}+\frac{1}{b+c}\right)+\frac{c}{4}\left(\frac{1}{a+c}+\frac{1}{b+c}\right)\)
\(\Rightarrow VT\le\frac{a}{4\left(a+b\right)}+\frac{a}{4\left(a+c\right)}+\frac{b}{4\left(a+b\right)}+\frac{b}{4\left(b+c\right)}+\frac{c}{4\left(a+c\right)}+\frac{c}{4\left(b+c\right)}\)
\(\Rightarrow VT\le\left[\frac{a}{4\left(a+b\right)}+\frac{b}{4\left(a+b\right)}\right]+\left[\frac{a}{4\left(a+c\right)}+\frac{c}{4\left(a+c\right)}\right]+\left[\frac{b}{4\left(b+c\right)}+\frac{c}{4\left(b+c\right)}\right]\)
\(\Rightarrow VT\le\frac{a+b}{4\left(a+b\right)}+\frac{a+c}{4\left(a+c\right)}+\frac{b+c}{4\left(b+c\right)}\)
\(\Rightarrow VT\le\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)
\(\Rightarrow VT\le\frac{3}{4}\)
\(\Leftrightarrow\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\le\frac{3}{4}\) ( đpcm )
