<s>[INST] Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$. Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$ for all notnegative value of $x,y,z$.
I think that minimum value is $\frac{3}{4}$ when $x=y=z$ but I have no prove.
 [/INST] $f(x;y;z)=\frac{(xy+yz+zx)(y-z)^2 + (xy+xz-2yz)^2}{4(x+y)(z+x)(y+z)^2} +\frac{3}{4} \geq \frac{3}{4}$
Equality holds when $x=y=z$
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