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

\(VT=\frac{2\left(yz\right)^2}{xy+xz}+\frac{2\left(zx\right)^2}{xy+yz}+\frac{2\left(xy\right)^2}{xz+yz}\)

\(VT\ge\frac{2\left(xy+yz+zx\right)^2}{2\left(xy+yz+zx\right)}=xy+yz+zx\)

Dấu "=" xảy ra khi \(x=y=z=\frac{1}{\sqrt[3]{2}}\)

Bài 1:Áp dụng C-S dạng engel

\(\frac{3}{xy+yz+xz}+\frac{2}{x^2+y^2+z^2}=\frac{6}{2\left(xy+yz+xz\right)}+\frac{2}{x^2+y^2+z^2}\)

\(\ge\frac{\left(\sqrt{6}+\sqrt{2}\right)^2}{\left(x+y+z\right)^2}=\left(\sqrt{6}+\sqrt{2}\right)^2>14\)

Ta có

\(x^2y^2+y^2z^2+z^2x^2\ge xyz\left(x+y+z\right)\)

\(=>x^2y^2+y^2z^2+z^2x^2+2\left(xyz\right)\left(x+y+z\right)\ge3xyz\left(x+y+z\right)\)

\(=>\left(xy+yz+zx\right)^2\ge3\left(x+y+z\right)\)

\(=>\frac{1}{\left(x+y+z\right)}\ge\frac{3}{\left(xy+yz+zx\right)^2}\)

\(=>A\ge\frac{3}{\left(xy+yz+zx\right)^2}-\frac{2}{xy+yz+zx}\)

đặt 

\(\frac{1}{xy+yz+zx}=t\)

\(=>A\ge3t^2-2t\)

mà \(\left(3t-1\right)^2\ge0=>9t^2-6t+1\ge0=>3t^2-2t+\frac{1}{3}\ge0\Rightarrow3t^2-2t\ge-\frac{1}{3}\)

\(=>A\ge-\frac{1}{3}\)(dpcm)

Dấu = xảy ra khi x=y=z=1

tinh tuoi con gai bang 1/4 tuoi me , tuoi con bang 1/5 tuoi me . tuoi con gai cong voi tuoi cua con trai 

la 18 tuoi . hoi me bao nhieu tuoi ?

r5464

a: =>x^2+y^2+z^2-4x+2y-6z+14=0

=>x^2-4x+4+y^2+2y+1+z^2-6z+9=0

=>(x-2)^2+(y+1)^2+(z-3)^2=0

=>x=2; y=-1; z=3

b: \(\left(x+y+z\right)\cdot\left(xy+yz+xz\right)\)

\(=x^2y+xyz+x^2z+xy^2+y^2z+xyz+xyz+yz^2+xz^2\)

\(=x^2y+xy^2+y^2z+x^2z+yz^2+xz^2+3xyz\)

Theo đề, ta có:

\(x^2y+xy^2+y^2z+x^2z+yz^2+xz^2+2xyz=0\)

\(\Leftrightarrow x^2y+2xyz+yz^2+xy^2+2xzy+xz^2+zx^2-2xyz+zy^2=0\)

\(\Leftrightarrow y\left(x+z\right)^2+x\left(y+z\right)^2+z\left(x+y\right)^2=0\)

=>x=y=z=0

=>x^2013+y^2013+z^2013=(x+y+z)^2013

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

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

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

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