Inequality involving $x^n$ This is probably not very exciting, but can't get my head around this for a while. Here's the inequality:
$$
x_1^3-\epsilon x_1^2 -(2+\epsilon(1-x_1^n))x_1+(1+\epsilon)>0
$$
where $\epsilon>0, n \in \mathbb{Z^{+}}$. Please just give me hints, don't solve it for me.
EDIT: $x_1>0$    
EDIT2: it seems that the solution to inequality $1-x_1-x_1^2>0$ is the trick. For (see EDIT 1) $0<x_1<\frac{\sqrt{5}-1}{2} \approx 0.618$ the main inequality above is true for all $n, \epsilon$. If $x_1$ is larger than $\approx 0.618$, then it is true for some $\epsilon>0$ iff $n$ is not very large. The condition $\epsilon>0$ is crucial.

If it is wished that the inequality be true for all positive $x_1$, it will need modification.  For example, let $n=1$. Then we are looking at 
$$x^3-(2+\epsilon)x+1+\epsilon.$$
This is not necessarily positive. Imagine $\epsilon$ very close to $0$. The function reaches a minimum at $x=\sqrt{(2+\epsilon)/3}$, and the minimum value is negative, though not by much. I chose a small $\epsilon$, because presumably that is what is intended. But if we pick $\epsilon$ large, like $10$, we are looking at $z^3-12x+11$, which is $-5$ at $x=2$.
There will be similar difficulties with $n=2$. And for any $n\ge 3$, and small $\epsilon$, we can reproduce the same problem. 
