<s>[INST] Given a polynomial find the minimum value of the variable. If $x^5 - x^3 + x = a. $
Then we have to find the minimum value of $x^6$ in terms of a.
The answer given is $2a - 1$ if that gives any idea.
I have no idea how to approach this problem. 
A hint would do fine.
 [/INST] I reinterprete the problem a bit differently (based upon the proposed answer) to show that
$$\tag1x^5-x^3+x=a\implies x^6\ge 2a-1. $$
Note that $(1)$ is trivially true for $a\le \frac12$. Hence we may assume that $a$ is positive.
We have $$(x^2+1)a=(x^2+1)(x^5-x^3+x)=x^7+x $$
so that $x$ must be positive. 
Divide by $x$ and subtract $1$ to arrive at 
$$x^6=\frac{(x^2+1)a}{x}-1\ge\frac{2xa}{x}-1=2a-1, $$
where we used $x>0$ and $x^2+1\ge 2x$ (from $x^2+1-2x=(x-1)^2\ge 0$).
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