[tex]\displaystyle\\
\sqrt{(x-1 )^{2} } + x^{100} +x \geq 1 \\\\
x\in [0;~1] ~~~\Longrightarrow~~~ \boxed{0 \leq x^{100} \leq 1}\\\\
\sqrt{(x-1)^2} = \Big|x-1\Big| = \boxed{1-x} ~~~\text{deoarece: }~~ x \leq 1\\\\
\text{Rezolvare:}\\\\
\sqrt{(x-1 )^{2} } + x^{100} +x = \\\\
= \Big|x-1\Big| + x^{100} + x = \\\\
= 1-x + x^{100} +x = \\\\
=1\underline{-x}+\underline{x} +x^{100} = \boxed{\bf \boxed{\bf 1 + x^{100} \geq 1}}[/tex]
