For $x,\,y,\,z>0$ and $x^2y^2z^2+\left(1+x\right)\left(1+y\right)\left(1+z\right)\ge x+y+z+xy+yz+zx+3.$ Find Min of:$$P=\dfrac{x^3}{\left(y+2z\right)\left(2z+3x\right)}+\dfrac{y^3}{\left(z+2x\right)\left(2x+3y\right)}+\dfrac{z^3}{\left(x+2y\right)\left(2y+3z\right)}$$
I don't know any ideas to solve for it, I think it is very difficult to me so I need your help and I will think about it, tks.
A stationary point is $x=y=z=1$ which gives $P=1/5$. I have not checked the second order sufficient conditions for a minimum, but having the candidate point you should be able to do it by yourself, in case I can help you further. For a reference you can look up for the KKT theorem.