\(c+ab=\left(a+b+c\right)c+ab=ac+cb+c^2+ab=\left(a+c\right)\left(b+c\right)\)

Tương tự : \(a+bc=\left(a+b\right)\left(a+c\right);c+ab=\left(c+a\right)\left(c+b\right)\)

\(P=\sqrt{\frac{ab}{\left(c+a\right)\left(c+b\right)}}+\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}+\sqrt{\frac{ca}{\left(b+c\right)\left(b+a\right)}}\)

áp dụng bất đẳng tức cauchy :

\(\sqrt{\frac{ab}{\left(c+a\right)\left(c+b\right)}}\le\frac{1}{2}\left(\frac{a}{c+a}+\frac{b}{c+b}\right)\)

\(\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\le\frac{1}{2}\left(\frac{b}{a+b}+\frac{c}{a+c}\right)\)

\(\sqrt{\frac{ca}{\left(b+c\right)\left(b+a\right)}}\le\frac{1}{2}\left(\frac{c}{b+c}+\frac{a}{b+a}\right)\)

cộng vế theo vế 

\(\Rightarrow P\le\frac{1}{2}\left(\frac{a}{a+c}+\frac{b}{c+b}+\frac{b}{a+b}+\frac{c}{a+c}+\frac{c}{b+c}+\frac{a}{b+a}\right)\)

\(\Leftrightarrow P\le\frac{1}{2}\left(\frac{a+c}{a+c}+\frac{b+c}{b+c}+\frac{a+b}{a+b}\right)=\frac{1}{2}\cdot3=\frac{3}{2}\)

dấu "=" xảy ra khi a=b=c=1/3

Có a+b+c=1 => c=(a+b+c).c=ac+bc+c²

\(\Rightarrow c+ab=ac+bc+c^2+ab=a\left(b+c\right)+c\left(b+c\right)=\left(b+c\right)\left(a+c\right)\)

\(\Rightarrow\sqrt{\frac{ab}{c+ab}}=\sqrt{\frac{ab}{\left(c+a\right)\left(c+b\right)}}\le\frac{\frac{a}{c+b}+\frac{b}{c+b}}{2}\)

Tương tự ta có \(\hept{\begin{cases}a+bc=\left(a+b\right)\left(a+c\right)\\b+ac=\left(b+a\right)\left(b+c\right)\end{cases}\Leftrightarrow\hept{\begin{cases}\sqrt{\frac{bc}{a+bc}}=\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\le\frac{\frac{b}{a+b}+\frac{c}{a+c}}{2}\\\sqrt{\frac{ca}{b+ca}}=\sqrt{\frac{ca}{\left(b+c\right)\left(b+a\right)}}\le\frac{\frac{c}{b+c}+\frac{a}{b+a}}{2}\end{cases}}}\)

\(\Rightarrow P\le\frac{\frac{b}{a+b}+\frac{c}{c+a}+\frac{c}{b+c}+\frac{a}{a+b}+\frac{a}{c+a}+\frac{b}{c+b}}{2}\)\(=\frac{\frac{a+c}{a+c}+\frac{c+b}{c+b}+\frac{a+b}{a+b}}{2}=\frac{3}{2}\)

Dấu "=" xảy ra khi \(a=b=c=\frac{1}{3}\)

tham khảo

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Theo BĐT \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\) ta có:

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

Tương tự ta cũng có các BĐT sau:

\(\frac{bc}{a+1}\le\frac{1}{4}\left(\frac{bc}{a+b}+\frac{bc}{a+c}\right);\frac{ca}{b+1}\le\frac{1}{4}\left(\frac{ac}{a+b}+\frac{ac}{b+c}\right)\)

Cộng theo vế các BĐT cùng dấu có:

\(Q\le\frac{1}{4}\left(\frac{c\left(a+b\right)}{a+b}+\frac{a\left(b+c\right)}{b+c}+\frac{b\left(c+a\right)}{c+a}\right)\)

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

Khi a=b=c=1/3

Ta có: \(a+b+c=1\Leftrightarrow a^2+ab+ca=a\)

Thay vào ta có: \(\sqrt{\frac{bc}{a+bc}}=\sqrt{\frac{bc}{a^2+ab+ca+bc}}=\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\)

Áp dụng Cauchy ngược: \(\sqrt{\frac{bc}{a+bc}}=\sqrt{\frac{bc}{a^2+ab+ca+bc}}\le\frac{\frac{b}{a+b}+\frac{c}{a+c}}{2}\)

Tương tự ta CM được: \(\sqrt{\frac{ab}{c+ab}}\le\frac{\frac{a}{c+a}+\frac{b}{c+b}}{2}\)

                                     \(\sqrt{\frac{ca}{b+ca}}\le\frac{\frac{c}{b+c}+\frac{a}{b+a}}{2}\)

Cộng vế 3 BĐT trên ta được:

\(P\le\frac{\frac{a}{c+a}+\frac{b}{c+b}+\frac{b}{a+b}+\frac{c}{a+c}+\frac{c}{b+c}+\frac{a}{b+a}}{2}\)

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

\(=\frac{1+1+1}{2}=\frac{3}{2}\)

Dấu "=" xảy ra khi: \(a=b=c=\frac{1}{3}\)

Vậy \(Max_P=\frac{3}{2}\Leftrightarrow a=b=c=\frac{1}{3}\)

Ta có :

\(c+ab=\left(a+b+c\right)c+ab=ac+ac+c^2+ab=\left(a+c\right)\left(b+c\right)\)

Tương tự :  \(a+bc=\left(a+b\right)\left(a+c\right);c+ab=\left(c+b\right)\left(c+a\right)\)

 \(\Rightarrow P=\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}+\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}+\sqrt{\frac{ca}{\left(c+a\right)\left(c+b\right)}}\)

Áp dụng BĐT cauchy :

\(\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}\le\frac{1}{2}\left(\frac{a}{a+c}+\frac{b}{b+c}\right)\)

\(\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\le\frac{1}{2}\left(\frac{b}{a+b}+\frac{c}{a+c}\right)\)

\(\sqrt{\frac{ca}{\left(c+b\right)\left(c+a\right)}}\le\frac{1}{2}\left(\frac{c}{c+b}+\frac{a}{c+a}\right)\)

Cộng vế với vế :

\(\Rightarrow P\le\frac{1}{2}\left(\frac{a}{a+c}+\frac{b}{b+c}+\frac{b}{a+b}+\frac{c}{a+c}+\frac{c}{c+b}+\frac{a}{c+a}\right)\)

\(\Leftrightarrow P\le\frac{1}{2}\left(\frac{a+c}{a+b}+\frac{b+c}{b+c}+\frac{a+b}{a+b}\right)=\frac{1}{2}.3=\frac{3}{2}\)

Dấu "=" xảy ra \(\Leftrightarrow a=b=c=\frac{1}{3}\)

có a+b+c=1 => c=(a+b+c).c=ac + bc + c²

=> c+ab=ac+bc+c²+ab=a(c+b)+c(b+c)=(c+a)(c+b)

=> \(\sqrt{\frac{ab}{c+ab}}=\sqrt{\frac{ab}{\left(a+c\right)\left(c+b\right)}}\le\frac{\frac{a}{c+a}+\frac{b}{b+c}}{2}\)

tương tự có \(\hept{\begin{cases}a+bc=\left(a+b\right)\left(a+c\right)\\b+ca=\left(b+c\right)\left(b+a\right)\end{cases}\Rightarrow\sqrt{\frac{bc}{a+bc}}=\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\le\frac{\frac{b}{a+b}+\frac{c}{a+c}}{2}}\)và \(\sqrt{\frac{ca}{b+ca}}=\sqrt{\frac{ca}{\left(b+c\right)\left(b+a\right)}}\le\frac{\frac{c}{b+c}+\frac{a}{b+a}}{2}\)

\(\Rightarrow P\le\frac{\frac{a}{c+a}+\frac{b}{b+c}+\frac{b}{a+b}+\frac{c}{c+a}+\frac{a}{a+b}}{2}=\frac{\frac{a+c}{a+c}+\frac{c+b}{c+b}+\frac{b+a}{b+a}}{2}=\frac{3}{2}\)

dấu "=" xảy ra khi \(a=b=c=\frac{1}{3}\)

vậy maxP=\(\frac{3}{2}\)đạt được khi a=b=c=\(\frac{1}{3}\)

 Ta có : \(\sqrt{\frac{ab}{ab+2c}}=\sqrt{\frac{ab}{ab+\left(a+b+c\right)c}}=\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}\le\frac{1}{2}\left(\frac{a}{a+c}+\frac{b}{b+c}\right)\)

Đẳng thức xảy ra khi và chỉ khi \(\frac{a}{a+c}+\frac{b}{b+c}\)

Tương tự ta cũng có 

           \(\sqrt{\frac{bc}{bc+2a}}\le\frac{1}{2}\left(\frac{b}{b+a}+\frac{c}{c+a}\right);\sqrt{\frac{ca}{ca+2b}}\le\frac{1}{2}\left(\frac{c}{c+a}+\frac{a}{a+b}\right)\)

Cộng các vế ta được \(S\le\frac{1}{2}\left(\frac{a+b}{a+b}+\frac{b+c}{b+c}+\frac{c+a}{c+a}\right)=\frac{3}{2}\)

Đẳng thức xảy ra khi và chỉ khi \(a=b=c=\frac{2}{3}\)

Vậy \(S_{max}=\frac{3}{2}\Leftrightarrow x=y=z=\frac{2}{3}\)

\(3-P=1-\frac{x}{x+1}+1-\frac{y}{y+1}+1-\frac{z}{z+1}\)

\(=\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\ge\frac{9}{x+y+z+3}=\frac{9}{1+3}=\frac{9}{4}\)

\(\Rightarrow P\le\frac{3}{4}\)

Dấu "=" xảy ra tại \(x=y=z=\frac{1}{3}\)

2/\(LHS\ge\frac{\left(a+b+c\right)^2}{a+b+c+\sqrt[3]{bc}+\sqrt[3]{ca}+\sqrt[3]{ab}}\)

\(\ge\frac{\left(a+b+c\right)^2}{a+b+c+\frac{1+b+c}{3}+\frac{1+c+a}{3}+\frac{1+a+b}{3}}=\frac{3}{2}\)

Bài 1 : 

\(P=\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}\)

Thay \(x+y+z=1\)vào biểu thức 

\(\Rightarrow P=\frac{x}{2x+y+z}+\frac{y}{x+2y+z}+\frac{z}{x+y+2z}\)

Áp dụng bất đẳng thức \(\frac{1}{a+b}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}\right)\forall a,b>0\)

\(\Rightarrow\hept{\begin{cases}\frac{x}{2x+y+z}=\frac{x}{x+y+x+z}\le\frac{x}{4}\left(\frac{1}{x+y}+\frac{1}{x+z}\right)\\\frac{y}{x+2y+z}=\frac{y}{x+y+y+z}\le\frac{y}{4}\left(\frac{1}{x+y}+\frac{1}{y+z}\right)\\\frac{z}{x+y+2z}=\frac{z}{x+z+y+z}\le\frac{z}{4}\left(\frac{1}{x+z}+\frac{1}{y+z}\right)\end{cases}}\)

\(\Rightarrow VT\le\frac{x}{4}\left(\frac{1}{x+y}+\frac{1}{x+z}\right)+\frac{y}{4}\left(\frac{1}{x+y}+\frac{1}{y+z}\right)\)

\(+\frac{z}{4}\left(\frac{1}{x+z}+\frac{1}{y+z}\right)\)

\(\Rightarrow VT\le\frac{x}{4\left(x+y\right)}+\frac{y}{4\left(x+y\right)}+\frac{x}{4\left(x+z\right)}+\frac{z}{4\left(x+z\right)}+\frac{y}{4\left(y+z\right)}\)

\(+\frac{z}{4\left(y+z\right)}\)

\(\Rightarrow VT\le\frac{x+y}{4\left(x+y\right)}+\frac{x+z}{4\left(x+z\right)}+\frac{y+z}{4\left(y+z\right)}\)

\(\Rightarrow VT\le\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\)

\(\Rightarrow P\le\frac{3}{4}\)

Vậy \(P_{max}=\frac{3}{4}\)

Dấu " = " xảy ra khi \(x=y=z\)

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