<s>[INST] If $x+y+z=1$ and $x,y,z>0\,$ find $\min xy(x+y)^2+yz(y+z)^2+zx(z+x)^2$ 
If $x+y+z=1$ and $x,y,z>0\,$ find the minimum of  $xy(x+y)^2+yz(y+z)^2+zx(z+x)^2$

$\bf{My\; Try::}$ We can write expression as $xy(1-z)^2+yz(1-x)^2+zx(1-y)^2$
$ = xy(1+z^2-2z)+yz(1+x^2-2x)+zx(1+y^2-2y)$
$ = xyz(x+y+z)+(xy+yz+zx)-6xyz= xy+yz+zx-5xyz$
Now how can I solve it after that, help required, thanks
 [/INST] The minimum does not exist, but arbitrarily small values can be reached by letting $x,y\to0$, $z\to1$; i.e., the infimum is $0$.
Explicitly, $x=y=\varepsilon$ and $z=1-2\varepsilon$ gives
$$xy(x+y)^2+yz(y+z)^2+zx(z+x)^2\leq3\times\varepsilon\cdot1\cdot(1+1)^2=12\varepsilon.$$
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