Đặt \(\hept{\begin{cases}2x+y+z=4a\\2y+x+z=4b\\2z+x+y=4c\end{cases}\Rightarrow}\hept{\begin{cases}x=3a-b-c\\y=3b-c-a\\z=3c-a-b\end{cases}}\)thay vào biểu thức đó

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

\(=\frac{3a-b-c}{4a}+\frac{3b-c-a}{4b}+\frac{3c-a-b}{4c}\)

\(=\frac{3}{4}-\frac{b-c}{4a}+\frac{3}{4}-\frac{c-a}{4b}+\frac{3}{4}-\frac{a-b}{4c}\)

\(=\frac{9}{4}-\frac{1}{4}\left(\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\right)\)

Áp dụng BĐT sau: \(\frac{a}{b}+\frac{b}{a}\ge2\Rightarrow\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\ge6\)

\(\Leftrightarrow\frac{1}{4}\left(\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\right)\ge\frac{6}{4}\)

\(\Leftrightarrow\frac{9}{4}-\frac{1}{4}\left(\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\right)\le\frac{3}{4}\)

Từ đó ta có: \(\frac{x}{2x+y+z}+\frac{y}{2y+x+z}+\frac{z}{2z+x+y}\le\frac{3}{4}\)(đpcm).

Dấu "=" xảy ra <=> x=y=z.

Bài của lớp 7 ghê vậy!!

Áp dụng bất đẳng thức Cauchy cho 3 số dương x,y,z

ta có bổ đề \((a+b+c)({1\over a}+{1\over b}+{1\over c})\)  >  9

Áp dụng vào ta có

\(D*({2x+y+z\over x}+{2y+x+z\over y}+{2z+y+x\over z})\)  >9(1)

Ta có \({2x+y+z\over x}+{2y+x+z\over y}+{2z+y+x\over z}\) =\(2+{y+z\over x}+2+{z+x\over y}+2+{y+x\over z}\)=\(6-3+{y+z\over x}+1+{z+x\over y}+1+{y+x\over z}+1\)=\(3+{x+y+z\over x}+{y+x+z\over y}+{z+y+x\over z}\)=\(3+(x+y+z)({1\over x}+{1\over y}+{1\over z})\)  >  3+9=12

thay vào(1)

Ta có \(D \) <  \({9\over 12}\)=\({3\over 4}\) 

Dấu "=" xảy ra khi x=y=z 

=> ĐPCM

áp dụng bất đẳng thức phụ : \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

\(\frac{x}{2x+y+z}=\frac{x}{x+y+x+z}\le\frac{1}{4}\left(\frac{x}{x+y}+\frac{x}{x+z}\right)\)

\(\frac{y}{2y+x+z}\le\frac{1}{4}\left(\frac{y}{y+x}+\frac{y}{y+z}\right)\)

\(\frac{z}{2z+x+y}\le\frac{1}{4}\left(\frac{1}{x+z}+\frac{1}{y+z}\right)\)

cộng vế theo vế

\(\frac{x}{2x+y+z}+\frac{y}{2y+z+x}+\frac{z}{2z+x+y}\le\frac{1}{4}\left(\frac{x+y}{x+y}+\frac{y+z}{y+z}+\frac{z+x}{z+x}\right)=\frac{1}{4}\cdot3=\frac{3}{4}\)(đpcm)

Đặt \(a=2x+y+z;b=2y+z+x;c=2z+x+y\)

\( \implies\) \(a+b+c=\left(2x+y+z\right)+\left(2y+z+x\right)+\left(2z+x+y\right)\) 

\( \implies\) \(a+b+c=4x+4y+4z\)

\( \implies\) \(x+y+z=\frac{a+b+c}{4}\) 

+)Ta có : \(a=2x+y+z\)

\(\iff\) \(a=x+\left(x+y+z\right)\)

\(\iff\) \(a-\left(x+y+z\right)=x\)

\(\iff\) \(a-\frac{a+b+c}{4}=x\)

\(\iff\) \(x=\frac{3a-b-c}{4}\)

+)Ta có :\(b=2y+z+x\)

\(\iff\) \(b=y+\left(y+z+x\right)\)

\(\iff\)\(b-\left(y+z+x\right)=y\)

\(\iff\) \(b-\frac{a+b+c}{4}=y\)

\(\iff\)\(y=\frac{3b-c-a}{4}\)

+)Ta có :\(c=2z+x+y\)

\(\iff\) \(c=z+\left(z+x+y\right)\)

\(\iff\) \(c-\left(z+x+y\right)=z\)

\(\iff\) \(c-\frac{a+b+c}{4}=z\)

\(\iff\)\(z=\frac{3c-a-b}{4}\)

​​​\( \implies\)​ \(\frac{x}{2x+y+z}+\frac{y}{2y+z+x}+\frac{z}{2z+x+y}\) 

 \(=\frac{3a-b-c}{4a}+\frac{3b-c-a}{4b}+\frac{3c-a-b}{4c}\)

 \(=\frac{9}{4}-\left(\frac{b}{4a}+\frac{c}{4a}+\frac{c}{4b}+\frac{a}{4b}+\frac{a}{4c}+\frac{b}{4c}\right)\)

 \(=\frac{9}{4}-\frac{1}{4}\left(\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\right)\)

 \(=\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\)

Áp dụng bất đẳng thức ( BĐT Cosi ) : \(m+n\)\( \geq\)\(2\sqrt{mn}\) \(\left(m;n>0\right)\)ta được : 

\(\frac{b}{a}+\frac{a}{b}\) \( \geq\) 2 \(\sqrt{\frac{b}{a}.\frac{a}{b}}\) = 2 \( \implies\) \(\frac{b}{a}+\frac{a}{b}\) \( \geq\) 2 

\(\frac{c}{a}+\frac{a}{c}\) \( \geq\) 2 \(\sqrt{\frac{c}{a}.\frac{a}{c}}\) = 2 \( \implies\) \(\frac{c}{a}+\frac{a}{c}\) \( \geq\) 2 

\(\frac{b}{c}+\frac{c}{b}\) \( \geq\) 2 \(\sqrt{\frac{b}{c}.\frac{c}{b}}\) = 2 \( \implies\) \(\frac{b}{c}+\frac{c}{b}\) \( \geq\) 2 

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

\( \implies\) ​​​\(\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\)​ \( \geq\) 6 

\( \implies\) \(\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \( \geq\) \(\frac{6}{4}\)

\( \implies\) \(\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \( \geq\) \(\frac{3}{2}\)

\( \implies\) \(-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(-\frac{3}{2}\)

\( \implies\) \(\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(\frac{9}{4}-\frac{3}{2}\)

\( \implies\) \(\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(\frac{3}{4}\) 

Dấu " = " xảy ra khi a = b = c hay x = y = z 

https://chat.lazi.vn/upload/images/2020/03/file_xot1584973258.jpg

Chúc bạn hok tốt!!!

\(\frac{x}{2x+y+z}=\frac{x}{\left(x+y\right)+\left(x+z\right)}\le\frac{1}{4}\left(\frac{x}{x+y}+\frac{x}{x+z}\right)\le\frac{1}{16}\left(\frac{x}{x}+\frac{x}{y}+\frac{x}{x}+\frac{x}{z}\right)=\frac{1}{16}\left(2+\frac{x}{y}+\frac{x}{z}\right)\)

\(tươngtự:\frac{y}{2y+z+x}\le\frac{1}{16}\left(2+\frac{y}{z}+\frac{y}{x}\right);\frac{z}{2z+x+y}\le\frac{1}{16}\left(2+\frac{z}{x}+\frac{z}{y}\right).\text{Cộng vế theo vế ta được:}\frac{x}{2x+y+z}+\frac{y}{2y+z+x}+\frac{z}{2z+y+x}\le\frac{1}{16}\left(2+2+2+\frac{x}{y}+\frac{y}{x}+\frac{z}{x}+\frac{x}{z}+\frac{y}{z}+\frac{z}{y}\right)=\frac{1}{16}\left[6+\left(\frac{x}{y}+\frac{y}{x}\right)+\left(\frac{z}{x}+\frac{x}{z}\right)+\left(\frac{y}{z}+\frac{z}{y}\right)\right]\ge\frac{1}{16}\left(6+2\sqrt{\frac{xy}{xy}}+2\sqrt{\frac{xz}{xz}}+2\sqrt{\frac{yz}{yz}}\right)=\)

\(=\frac{12}{16}=\frac{3}{4}\Rightarrow\frac{x}{2x+y+z}+\frac{y}{2y+z+x}+\frac{z}{2z+x+y}\le\frac{3}{4}\left(\text{đpcm}\right)\)

dự đoán điểm rơi : x = y = z > 0 

dùng Cô si :bớt. Ta làm như sau 

Giải :    Đặt P = \(\frac{x^3}{y\left(z+2x\right)}+\frac{y^3}{z\left(x+2y\right)}+\frac{z^3}{x\left(y+2z\right)}.\)áp dụng bất đẳng thức Cô si cho 3 số dương.

ta có :  \(\frac{x^3}{y\left(z+2x\right)}+\frac{y}{3}+\frac{z+2x}{9}\ge3\sqrt[3]{\frac{x^3.y.\left(z+2x\right)}{y.\left(z+2x\right).3.9}}=x.\left(1\right)..\)

chứng minh tương tự ta có :

\(\frac{y^3}{z.\left(x+2y\right)}+\frac{z}{3}+\frac{x+2y}{9}\ge y\left(2\right).\)\(\frac{z^3}{x.\left(y+2z\right)}+\frac{x}{3}+\frac{y+2z}{9}\ge z.\left(3\right).\)

Cộng vế với vế các bất đẳng thức (1) , (2) và (3) ta được :  

                                 \(P+\frac{2}{3}.\left(x+y+z\right)\ge x+y+z\)

                =>    \(P\ge\frac{x+y+z}{3}.\) đấu " = " xẩy ra khi x = y = z  > 0  ( đpcm ) 

Cậu vào đây nha ! 

Câu hỏi của doanthihuong - Toán lớp 7 - Học toán với OnlineMath

Áp dụng bđt Cauchy-Schwarz:

\(\frac{x}{2x+y+z}+\frac{y}{2y+x+z}+\frac{z}{2z+x+y}\)

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

\(\le\frac{1}{4}\left(\frac{x}{x+y}+\frac{x}{x+z}+\frac{y}{x+y}+\frac{y}{y+z}+\frac{z}{y+z}+\frac{z}{x+z}\right)=\frac{3}{4}\)

\("="\Leftrightarrow x=y=z\)