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Ta có \(1+x^2=x^2+xy+yz+xz=\left(x+y\right)\left(x+z\right)\)

Tương tự  \(1+y^2=\left(x+y\right)\left(y+z\right)\)

\(1+z^2=\left(x+z\right)\left(y+z\right)\)

Thay vào A ta được

\(P=x\sqrt{\left(y+z\right)^2}+y\sqrt{\left(x+z\right)^2}+z\sqrt{\left(x+y\right)^2}\)

=2(xy+xz+yz)=2

17 tháng 6 2019

\(b,VT=VP\)

\(\Leftrightarrow\frac{x}{xy+yz+zx+x^2}+\frac{y}{xy+yz+zx+y^2}+\frac{z}{xy+yz+zx+z^2}\)

                                                                                                                                                                                                                                                                                    \(=\frac{2xyz}{\sqrt{\left(xy+yz+zx+x^2\right)\left(xy+yz+zx+y^2\right)\left(xy+yz+zx+z^2\right)}}\)

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

                                                                                \(=\frac{2xyz}{\sqrt{\left(x+y\right)\left(x+z\right)\left(y+z\right)\left(y+x\right)\left(z+x\right)\left(y+z\right)}}\)

\(\Leftrightarrow\frac{x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}=\frac{2xyz}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)

\(\Leftrightarrow xy+xz+xy+yz+xz+yz=2xyz\)

\(\Leftrightarrow2=2xyz\)

\(\Leftrightarrow xyz=1\)

Đù =)))

24 tháng 8 2019

Có xy + yz + zx = 1

=> 1 + x2 = x2 + xy + yz + zx

     1 + x2 = (x + y)(y + z)

Tương tự ta có: 

     1 + y2 = (y + x)(y + z)

     1 + z2 = (z + x)(z + y)

Thay vào P, ta được:

\(P=x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)\)

\(P=xy+yz+zx+xy+yz+zx\)

\(P=2\left(xy+yz+zx\right)=2\)

Vậy P = 2

18 tháng 10 2018

Xét hạng tử: \(x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{1+x^2}}\)

Thay \(xy+yz+zx=1\); ta có:

\(x\sqrt{\frac{\left(y^2+xy+yz+zx\right)\left(z^2+xy+yz+zx\right)}{x^2+xy+yz+zx}}=x\sqrt{\frac{\left(x+y\right)\left(y+z\right)^2\left(x+z\right)}{\left(x+y\right)\left(x+z\right)}}\)

\(=x\sqrt{\left(y+z\right)^2}=xy+xz\)

Tượng tự: \(y\sqrt{\frac{\left(1+z^2\right)\left(1+x^2\right)}{1+y^2}}=xy+yz;z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{1+z^2}}=xz+yz\)

Do đó: \(A=2\left(xy+yz+zx\right)=2.1=2\)

ĐS:...

17 tháng 5 2018

a) Ta có : \(1+x^2=xy+yz+zx+x^2=x\left(x+y\right)+z\left(x+y\right)=\left(x+y\right)\left(z+x\right)\)

b) \(\Sigma\left(x\sqrt{\dfrac{\left(1+y^2\right)\left(1+z^2\right)}{1+x^2}}\right)=\Sigma\left(x\sqrt{\dfrac{\left(x+y\right)\left(y+z\right).\left(x+z\right)\left(y+z\right)}{\left(x+y\right)\left(x+z\right)}}\right)\)

\(=\Sigma\left(x\left(y+z\right)\right)=xy+xz+xy+yz+zx+zy=2\left(xy+yz+zx\right)=2\)

17 tháng 8 2019

+ \(x\sqrt{\frac{\left(y^2+1\right)\left(z^2+1\right)}{x^2+1}}=x\sqrt{\frac{\left(xy+yz+zx+y^2\right)\left(xy+yz+zx+z^2\right)}{x^2+xy+yz+zx}}\)

\(=x\sqrt{\frac{\left(x+y\right)\left(y+z\right)\left(x+z\right)\left(y+z\right)}{\left(x+y\right)\left(x+z\right)}}=x\sqrt{\left(y+z\right)^2}=xy+xz\)

+ Tương tự : \(y\sqrt{\frac{\left(z^2+1\right)\left(x^2+1\right)}{y^2+1}}=xy+yz\)

\(z\sqrt{\frac{\left(x^2+1\right)\left(y^2+1\right)}{z^2+1}}=xz+yz\)

Do đó : \(P=2\left(xy+yz+zx\right)=2\)

14 tháng 9 2018

Bài này hình như x,y,z>0

Ta có: \(x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{1+x^2}}=x\sqrt{\frac{\left(y^2+xy+yz+zx\right)\left(z^2+xy+yz+zx\right)}{\left(x^2+xy+yz+zx\right)}}=x\sqrt{\frac{\left(y+x\right)\left(y+z\right)\left(z+x\right)\left(z+y\right)}{\left(x+y\right)\left(x+z\right)}}=x\sqrt{\left(y+z\right)^2}\)

Tương tự: \(y\sqrt{\frac{\left(1+z^2\right)\left(1+x^2\right)}{1+y^2}}=y\sqrt{\left(x+z\right)^2}\) 

                \(z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{1+z^2}}=z\sqrt{\left(x+y\right)^2}\)

Cộng từng vế, ta có: 

\(A=x\left(y+z\right)+y\left(z+x\right)+z\left(x+y\right)\) 

\(\Leftrightarrow A=2\left(xy+yz+zx\right)=2\)

14 tháng 9 2018

\(\hept{\begin{cases}1+y^2=y^2+xy+yz+zx=\left(x+y\right)\left(y+z\right)\\1+z^2=\left(z+x\right).\left(z+y\right)\\1+x^2=\left(x+y\right)\left(x+z\right)\end{cases}}\)

Thế vào \(A=x\sqrt{\left(y+z\right)^2}+y\sqrt{\left(x+z\right)^2}+z\sqrt{\left(x+y\right)^2}\)

\(=x\left|y+z\right|+y\left|x+z\right|+z\left|x+y\right|\)

\(=2\left(\left|xy\right|+\left|yz\right|+\left|zx\right|\right)\)

Nếu x,y,z\(\ge0\Rightarrow A=2\)

Nếu x,y,z\(< 0\)\(\Rightarrow A=-2\)

20 tháng 10 2020

1111111111111111111

\(VT=\Sigma\frac{xy+yz+zx}{xy}=3+\Sigma\frac{z\left(x+y\right)}{xy}\)

Đến đây để ý \(\frac{1}{2}\left[\frac{z\left(x+y\right)}{xy}+\frac{y\left(z+x\right)}{zx}\right]\ge\sqrt{\frac{\left(z+x\right)\left(x+y\right)}{x^2}}\left(\text{AM - GM}\right)\)

Là xong.

22 tháng 10 2017

ta có: xy+yz+zx=1

=> \(1+x^2=x^2+xy+yz+xz=\left(x+z\right)\left(x+y\right)\)

c/m tương tự ta đc: \(1+y^2=\left(x+y\right)\left(y+z\right)\)

                                \(1+z^2=\left(y+z\right)\left(z+x\right)\)

thay vào A ta đc:

\(A=x\sqrt{\frac{\left(x+y\right)\left(y+z\right)\left(y+z\right)\left(z+x\right)}{\left(x+z\right)\left(x+y\right)}}+y\sqrt{\frac{\left(y+z\right)\left(z+x\right)\left(x+z\right)\left(x+y\right)}{\left(x+y\right)\left(y+z\right)}}+z\sqrt{\frac{\left(x+z\right)\left(x+y\right)\left(x+y\right)\left(y+z\right)}{\left(y+z\right)\left(x+z\right)}}\)\(\Rightarrow A=x\sqrt{\left(y+z\right)^2}+y\sqrt{\left(x+z\right)^2}+z\sqrt{\left(x+y\right)^2}\)

\(\Rightarrow A=x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)\)

\(\Rightarrow A=2\left(xy+yz+zx\right)\)

\(\Rightarrow A=2\) vì xy+yz+zx=1

27 tháng 3 2019

Thay \(xy+yz+xz=1\) ta có: \(\hept{\begin{cases}1+x^2=xy+yz+xz+x^2=\left(x+z\right)\left(x+y\right)\\1+y^2=xy+yz+xz+y^2=\left(x+y\right)\left(y+z\right)\\1+z^2=xy+yz+xz+z^2=\left(x+z\right)\left(y+z\right)\end{cases}}\)

\(\Rightarrow S=x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)=2\left(xy+yz+xz\right)=2\)

25 tháng 9 2018

Ta co: \(1+x^2=xy+yz+zx+x^2=\left(x+y\right)\left(x+z\right)\)

\(\Rightarrow\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{\left(1+x^2\right)}}=\sqrt{\frac{\left(y+x\right)\left(y+z\right)\left(z+x\right)\left(z+y\right)}{\left(x+y\right)\left(x+z\right)}}=y+z\)

Thê vào ta được

\(A=x\left(y+z\right)+y\left(z+x\right)+z\left(x+y\right)=2\left(xy+yz+zx\right)=2\)