<s>[INST] Show that $x^{12}-x^9+x^4-x+1\geq0$ for all $x$ Show that $x^{12}-x^9+x^4-x+1\geq0$ for all $x$.  
When $x\leq0$ , this is easy.
When $x\geq1$, then also this is easy.
I need help with the case when $0\leq x\leq 1$
 [/INST] $x^{12} - x^9 +x^4 - x + 1 \ge 0$;
Rewritten:
$x^{12} + (x^4 - x^9) + (1 - x) \ge 0$.
For   $0 \le x \le 1$:
We have $x^{12} \ge 0$;   $(x^4 -x^9) \ge 0$ and  $(1-x) \ge 0$.
The sum of the above terms is $ \ge 0$.
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