<s>[INST] Prove this inequality with $xyz=1$ let $x,y,z>0$ and such $xyz=1$,show that
$$f(x)+f(y)+f(z)\le\dfrac{1}{8}$$
where $f(x)=\dfrac{x}{2x^{x+1}+11x^2+10x+1}$
I try use this $2x^x\ge x^2+1$,so we have
$$2x^{x+1}+11x^2+10x+1\ge x^3+11x^2+11x+1=(x+1)(x^2+10+1)$$
It need to prove
$$\sum_{cyc}\dfrac{x}{(x+1)(x^2+10x+1)}\le\dfrac{1}{8},$$where $xyz=1$
then I can't 
 [/INST] Now, let $x=\frac{a}{b},$ $y=\frac{b}{c}$, where $a$, $b$ and $c$ are positives.
Thus, $z=\frac{c}{a}$ and since $$x^x\geq x,$$ it's enough to prove that:
$$\sum_{cyc}\frac{ab}{13a^2+10ab+b^2}\leq\frac{1}{8},$$ 
which is true by BW.
Indeed, let $a=\min\{a,b,c\}$, $b=a+u$ and $c=a+v$.
Thus, we need to prove that:
$$384(u^2-uv+v^2)a^4+192(2u^3+7u^2v-5uv^2+2v^3)a^3+$$
$$+16(13u^4+82u^3v+39u^2v^2-62uv^3+13v^4)a^2+$$
$$+4(91u^3+258u^2v-162uv^2+13v^3)uva+13(13u^2+2uv+v^2)u^2v^2\geq0.$$
Now, we can show that:
$$384(u^2-uv+v^2)\geq384uv,$$
$$192(2u^3+7u^2v-5uv^2+2v^3)\geq768\sqrt{u^3v^3},$$
$$16(13u^4+82u^3v+39u^2v^2-62uv^3+13v^4)\geq-32u^2v^2,$$
$$4(91u^3+258u^2v-162uv^2+13v^3)\geq-384\sqrt{u^3v^3}$$ and 
$$13(13u^2+2uv+v^2)\geq112uv.$$
Now, let $a=t\sqrt{uv}.$
Thus, it's enough to prove that:
$$384t^4+768t^3-32t^2-384t+112\geq0,$$ which is smooth.
Can you end it now? 
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