<s>[INST] How to find the minimum value of $\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}$? Let $x,y,z\in [1,4]$ such that $x \geq y$ and $x \geq z$.
Find the minimum value of this expression:
$$
P=\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}
$$
 [/INST] As has been mentioned in comments, we want to find the minimum value of
$$
P=\frac{1}{2+3u}+\frac{u}{u+v}+\frac{v}{v+1}
$$
for $u,v\in[\frac{1}{4},1]$. Take partials of $P$ with respect to $u$ and $v$:
$$
\begin{align}
\frac{\partial P}{\partial u}&=-\frac{3}{(2+3u)^2}+\frac{v}{(u+v)^2}\\
\frac{\partial P}{\partial v}&=-\frac{u}{(u+v)^2}+\frac{1}{(v+1)^2}
\end{align}
$$
To find an interior extremum, we need $\frac{\partial P}{\partial u}=\frac{\partial P}{\partial v}=0$. In that case, we need
$$
0=u\frac{\partial P}{\partial u}+v\frac{\partial P}{\partial v}=-\frac{3u}{(2+3u)^2}+\frac{v}{(v+1)^2}
$$
However, for $u,v\in[\frac{1}{4},1]$, we have
$$
\begin{array}{c}
\frac{12}{121}\le\frac{3u}{(2+3u)^2}\le\frac{3}{25}\\
\frac{4}{25}\le\frac{v}{(v+1)^2}\le\frac{1}{4}
\end{array}
$$
Thus, $u\frac{\partial P}{\partial u}+v\frac{\partial P}{\partial v}\ge\frac{1}{25}$, so there can be no interior extremum.
Because $(u,v)\cdot\nabla P=u\frac{\partial P}{\partial u}+v\frac{\partial P}{\partial v}\ge\frac{1}{25}$ everywhere in $[\frac{1}{4},1]\times[\frac{1}{4},1]$, the minimum must be taken on the left or bottom edge of that square; i.e. $u=\frac{1}{4}$ or $v=\frac{1}{4}$.

$u=\frac{1}{4}$: $\frac{\partial P}{\partial v}=-\frac{4}{(1+4v)^2}+\frac{1}{(v+1)^2}=\frac{12v^2-3}{(1+4v)^2(v+1)^2}$ which vanishes at $v=\frac{1}{2}$.

Because $P(\frac{1}{4},\frac{1}{4})=\frac{117}{110}$ and $P(\frac{1}{4},\frac{1}{2})=\frac{34}{33}$ and $P(\frac{1}{4},1)=\frac{117}{110}$, $P(\frac{1}{4},\frac{1}{2})$ is a local minimum.

$v=\frac{1}{4}$: $\frac{\partial P}{\partial u}=-\frac{3}{(2+3u)^2}+\frac{4}{(4u+1)^2}=\frac{-12u^2+24u+13}{(2+3u)^2(4u+1)^2}$ which does not vanish on $[\frac{1}{4},1]$.

Because $P(\frac{1}{4},\frac{1}{4})=\frac{117}{110}$ and $P(1,\frac{1}{4})=\frac{33}{20}$, $P(\frac{1}{4},\frac{1}{4})$ is a local minimum.

Thus, $P(\frac{1}{4},\frac{1}{2})=\frac{34}{33}$ is the minimum of P for $(u,v)\in[\frac{1}{4},1]\times[\frac{1}{4},1]$.
</s>