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\(M=\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}\)
Vì xyz=1 nên \(x\ne0;y\ne0;z\ne0\)
Ta có \(\frac{1}{1+x+xy}=\frac{z}{\left(1+y+yz\right)xz}=\frac{xz}{z+xz+1}\)
Tương tự \(\frac{1}{1+y+yz}=\frac{xz}{\left(1+y+yz\right)xz}=\frac{xz}{xz+z+1}\)
Khi đó \(M=\frac{z}{z+xz+1}+\frac{xz}{xz+1+z}+\frac{1}{1+z+xz}=\frac{z+xz+1}{z+zx+1}=1\)
ta có x/xy+x+1 +y/yz+y+1 +z/xz+z+1
=xz/xyz+xz+z +xyz/xyz^2+xyz+xz +z/xz+z+1
=xz/1+xz+z +1/z+1+xz +z/ xz+z+1
=xz+z+1 /xz+z+1 =1
\(x+y+z=7\Rightarrow z=7-x-y\Rightarrow xy+z-6=xy+7-x-y-6=xy-x-y+1\)
\(=\left(x-1\right)\left(y-1\right)\)
Tương tự: \(yz+x-6=\left(y-1\right)\left(z-1\right);zx+y-6=\left(z-1\right)\left(x-1\right)\)
Viết lại: \(H=\frac{1}{\left(x-1\right)\left(y-1\right)}+\frac{1}{\left(y-1\right)\left(z-1\right)}+\frac{1}{\left(z-1\right)\left(x-1\right)}\)
\(=\frac{x-1+y-1+z-1}{\left(x-1\right)\left(y-1\right)\left(z-1\right)}=\frac{x+y+z-3}{xyz-\left(xy+yz+zx\right)+x+y+z-1}\)
\(=\frac{7-3}{3-13+7-1}=-1\)(Từ gt tính được \(xy+yz+zx=13\))
Ta có :
\(xy+yz+zx\)= \(\frac{\left(x+y+z\right)^2-x^2-y^2-z^2}{2}\)= \(\frac{7^2-23}{2}\)= \(13\)
Ta lại có :
\(xy+z-6=xy+z+1-x-y-z\)= \(\left(x-1\right)\left(y-1\right)\)
\(\Rightarrow A=\)\(\frac{1}{\left(x-1\right)\left(y-1\right)}\)\(+\)\(\frac{1}{\left(y-1\right)\left(z-1\right)}\)\(+\)\(\frac{1}{\left(z-1\right)\left(x-1\right)}\)
\(=\)\(\frac{x+y+z-3}{xyz-xy-yz-zx+x+y+z-1}\)
\(=-1\)
Ta có:
\(xy+yz+zx=\frac{\left(x+y+z\right)^2-x^2-y^2-z^2}{2}=\frac{7^2-23}{2}=13\)
Ta lại có:
\(xy+z-6=xy+z+1-x-y-z=\left(x-1\right)\left(y-1\right)\)
\(\Rightarrow A=\frac{1}{\left(x-1\right)\left(y-1\right)}+\frac{1}{\left(y-1\right)\left(z-1\right)}+\frac{1}{\left(z-1\right)\left(x-1\right)}\)
\(=\frac{x+y+z-3}{xyz-xy-yz-zx+x+y+z-1}=-1\)
Từ \(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}=1\)
\(\Rightarrow\)\(x+y+z=xyz\)
Ta có : \(\sqrt{yz\left(1+x^2\right)}=\sqrt{yz+x^2yz}=\sqrt{yz+x\left(x+y+z\right)}=\sqrt{\left(x+y\right)\left(x+z\right)}\)
Tương tự : \(\sqrt{xy\left(1+z^2\right)}=\sqrt{\left(z+y\right)\left(z+x\right)}\); \(\sqrt{zx\left(1+y^2\right)}=\sqrt{\left(y+z\right)\left(y+x\right)}\)
Nên \(Q=\frac{x}{\sqrt{\left(x+y\right)\left(x+z\right)}}+\frac{y}{\sqrt{\left(y+z\right)\left(y+x\right)}}+\frac{z}{\sqrt{\left(z+x\right)\left(z+y\right)}}\)
\(Q=\sqrt{\frac{x}{x+y}.\frac{x}{x+z}}+\sqrt{\frac{y}{x+y}.\frac{y}{y+z}}+\sqrt{\frac{z}{x+z}.\frac{z}{y+z}}\)
Áp dụng BĐT \(\sqrt{A.B}\le\frac{A+B}{2}\left(A,B>0\right)\)
Dấu "=" xảy ra khi A = B :
Ta được :
\(Q\le\frac{1}{2}\left(\frac{x}{x+y}+\frac{x}{x+z}+\frac{y}{y+x}+\frac{y}{y+z}+\frac{z}{z+x}+\frac{z}{z+y}\right)=\frac{3}{2}\)
Vậy GTLN của \(Q=\frac{3}{2}\)khi \(x=y=z=\sqrt{3}\)
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=0\Rightarrow\frac{x+y+z}{xyz}=0\Rightarrow x+y+z=0\Rightarrow x^3+y^3+z^3=3xyz\)
\(N=\frac{x^2}{yz}+\frac{y^2}{zx}+\frac{z^2}{xy}=\frac{x^3+y^3+z^3}{xyz}=\frac{3xyz}{xyz}=3\)
Có: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
=>\(\frac{yz+zx+xy}{xyz}=0\Rightarrow xy+yz+zx=0\)
\(P=\frac{yz}{x^2}+\frac{zx}{y^2}+\frac{xy}{z^2}=\frac{y^2z^2yz+z^2x^2xz+x^2y^2xy}{x^2y^2z^2}=\frac{\left(yz\right)^3+\left(zx\right)^3+\left(xy\right)^3}{x^2y^2z^2}\)
Ta có: nếu a+b+c=0 thì a^3 +b^3 +c^3 =3abc
Mà xy+yz+zx=0
=>\(\left(xy\right)^3+\left(yz\right)^3+\left(zx\right)^3=3\cdot xy\cdot yz\cdot zx=3x^2y^2z^2\)
=>\(P=\frac{3x^2y^2z^2}{x^2y^2z^2}=3\)
Cho mik sưa chút
\(P=\frac{xyz}{x^3}+\frac{xyz}{y^3}+\frac{xyz}{z^3}=xyz\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)\)
Áp dụng hằng đẳng thức a³ + b³ + c³ = [(a + b + c)(a² + b²+ c²-ab-bc-ca)+3abc]
\(\Rightarrow P=xyz\left[\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-\frac{1}{xy}-\frac{1}{yz}-\frac{1}{zx}\right)+3xyz\right]\)
\(\Rightarrow P=xyz.3.\frac{1}{xyz}=3\)