$f(x)=x^8-x^5+x^2-x+1$.
Prove that $f(x)>0$, for all $x \in\Bbb R$
I tried by calculating first derivative to find maxima or minima but it didn't worked.
2026-03-25 09:13:59.1774430039
Inequalities in functions
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Hint:
It amounts to showing that, for an $x$, $$x^8+x^2+1 >x^5+x.$$ If $x\le 0 $, this is obvious. If $x > 0$, consider the cases $x\ge 1$ and $x<1$. Can you show that each term in the r.h.s. is individually less than another term in the l.h.s.?