<s>[INST] How find the maximum of the value $t$ such $x^3+y^3+z^3+t(xy^2+yz^2+zx^2)\ge (1+t)(x^2y+y^2z+z^2x)$ let $x,y,z\ge 0$, find the maximum of the $t$, such 
$$x^3+y^3+z^3+t(xy^2+yz^2+zx^2)\ge (1+t)(x^2y+y^2z+z^2x)$$
maybe this take $y=x+u,z=x+t$,But I can't get the answer,because this is very ugly, Thanks
 [/INST] I assumed that it means that we need to find a maximal value of $t$, for which this inequality is true for any non-negatives $x$, $y$ and $z$.
If so, we can say that your way (BW) helps!
Let $x=\min\{x,y,z\}$, $y=x+u$, $z=x+v$ and $u=kv$.
Thus, $u$ and $v$ are non-negatives and $$\sum_{cyc}(x^2+txy^2-(t+1)x^2y)=$$
$$=2(u^2-uv+v^2)x+u^3-(t+1)u^2v+tuv^2+v^3\geq$$
$$\geq u^3-(t+1)u^2v+tuv^2+v^3=v^3(k^3-(t+1)k^2+tk+1).$$
Id est, it's enough to find a maximal value of $t$, for which the inequality $$k^3-(t+1)k^2+tk+1\geq0$$ is true for any non-negative value of $k$ or
$$k^3-k^2+1\geq t(k^2-k).$$
Now, it's enough to assume $k>1$ and it's remains to find$$\min_{k>1}\frac{k^3-k^2+1}{k^2-k}.$$
Can you end it now?
I got $$t_{max}=\frac{\sqrt{13+16\sqrt2}-1}{2}\approx2.484...$$
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