<s>[INST] Does a maximum value exist for this expression? Let $x$, $y$, $z$ be positive real numbers and $x + y + z =3$.  Does a maximum value exist for this expression? 
$$\displaystyle E = \frac{x}{2 y+3 z}+\frac{4 y}{5 z + 6 x}+\frac{7 z}{8x+9 y}.$$
I tried
Put 
$$a=2 y+3 z,\quad b=6 x+5 z,\quad c=8 x+9y.$$
Then
$$x = -\dfrac{45a}{242}+\dfrac{27b}{242}+\dfrac{5c}{121},\quad y = \dfrac{20a}{121}-\dfrac{12b}{121}+\dfrac{9c}{121}, \quad z = \dfrac{27a}{121}+\dfrac{8b}{121}-\dfrac{6c}{121}.$$
The expression $E$ has the form
$$E = \frac{378 a^2 b+160 a^2 c+112a b^2-225 a b c+72 a c^2+27b^2 c+10 b c^2}{242 a b c}.$$
From here, I can't solve it.
 [/INST] The expression $E$ is not bounded above. Let $x$ be very large, and $y$ and $z$ small. 
Edit: The condition $x+y+z=3$ has been added. The expression $E$ is still not bounded above. Let $x$ be very close to $3$, making $y$ and $z$ close to $0$. The term $\frac{x}{2y+3z}$ can be made arbitrarily large. 
</s>