cho A = \(\frac{xy+2x+1}{xy+x+y+1}+\frac{yz+2y+1}{yz+y+z+1}+\frac{zx+2z+1}{zx+z+x+1}\)với x,y,z\(\ne\)-1
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28 tháng 8 2018
\(A=\frac{xy+2y+1}{xy+x+y+1}+\frac{yz+2z+1}{yz+y+z+1}+\frac{zx+2x+1}{zx+z+x+1}\)
\(=\frac{y\left(x+1\right)+y+1}{\left(x+1\right)\left(y+1\right)}+\frac{z\left(y+1\right)+z+1}{\left(y+1\right)\left(z+1\right)}+\frac{x\left(z+1\right)+x+1}{\left(z+1\right)\left(x+1\right)}\)
\(=\frac{y}{y+1}+\frac{1}{x+1}+\frac{z}{z+1}+\frac{1}{y+1}+\frac{x}{x+1}+\frac{1}{z+1}\)
\(=\frac{y+1}{y+1}+\frac{z+1}{z+1}+\frac{x+1}{x+1}=3\)
NV
Nguyễn Việt Lâm
Giáo viên
23 tháng 10 2020
BĐT tương đương:
\(\frac{1}{z\left(1+\frac{1}{x}\right)}+\frac{1}{x\left(1+\frac{1}{y}\right)}+\frac{1}{y\left(1+\frac{1}{z}\right)}\ge2\)
Từ giả thiết:
\(xy+yz+zx+2xyz=1\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+2=\frac{1}{xyz}\)
Đặt \(\left(\frac{1}{x};\frac{1}{y};\frac{1}{z}\right)=\left(a;b;c\right)\Rightarrow a+b+c+2=abc\)
\(\Rightarrow a+b+c+2\le\frac{1}{27}\left(a+b+c\right)^3\)
\(\Leftrightarrow\left(a+b+c\right)^3-27\left(a+b+c\right)-54\ge0\)
\(\Leftrightarrow\left(a+b+c-6\right)\left(a+b+c+3\right)^2\ge0\)
\(\Leftrightarrow a+b+c\ge6\)
BĐT trở thành: \(\frac{c}{1+a}+\frac{a}{1+b}+\frac{b}{1+c}\ge2\)
Thật vậy, ta có:
\(VT=\frac{a^2}{a+ab}+\frac{b^2}{b+bc}+\frac{c^2}{c+ca}\ge\frac{\left(a+b+c\right)^2}{a+b+c+ab+bc+ca}\ge\frac{3\left(a+b+c\right)^2}{3\left(a+b+c\right)+\left(a+b+c\right)^2}\)
\(VT\ge\frac{3\left(a+b+c\right)}{3+a+b+c}=\frac{2\left(a+b+c\right)+a+b+c}{a+b+c+3}\ge\frac{2\left(a+b+c\right)+6}{a+b+c+3}=2\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c=2\) hay \(x=y=z=\frac{1}{2}\)
25 tháng 6 2016
Sửa lại đề là x;y;z khác -1.
\(A=\frac{xy+2x+1}{xy+x+y+1}+\frac{yz+2y+1}{yz+y+z+1}+\frac{zx+2z+1}{zx+z+x+1}=\)
\(A=\frac{x\left(y+1\right)+x+1}{x\left(y+1\right)+y+1}+\frac{y\left(z+1\right)+y+1}{y\left(z+1\right)+z+1}+\frac{z\left(x+1\right)+z+1}{z\left(x+1\right)+x+1}=\)
\(A=\frac{x\left(y+1\right)+x+1}{\left(x+1\right)\left(y+1\right)}+\frac{y\left(z+1\right)+y+1}{\left(y+1\right)\left(z+1\right)}+\frac{z\left(x+1\right)+z+1}{\left(z+1\right)\left(x+1\right)}=\)vì x;y;z khác -1 nên:
\(A=\frac{x}{x+1}+\frac{1}{y+1}+\frac{y}{y+1}+\frac{1}{z+1}+\frac{z}{z+1}+\frac{1}{x+1}=\)
\(A=\frac{x}{x+1}+\frac{1}{x+1}+\frac{y}{y+1}+\frac{1}{y+1}+\frac{z}{z+1}+\frac{1}{z+1}=\frac{x+1}{x+1}+\frac{y+1}{y+1}+\frac{z+1}{z+1}=1+1+1=3\)
A = 3 với mọi x;y;z khác -1 nên A không phụ thuộc vào x;y;z. đpcm
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