<s>[INST] Prove $\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}$ for $0 < x < 1$ I stumbled upon this question  while doing practice inequalities questions, and I do not know how to start...
Problem: Prove that 
\begin{align*}
\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}
\end{align*}
for $0 < x < 1$.
I thought possibly of having an intermediate equality, for example 
\begin{align*}
\sqrt{\frac{2x^2-2x+1}{2}}\ge\text{something}\ge\frac{1}{x+\frac{1}{x}}
\end{align*}
where the "something" is simple, but I could not deduce anything...any help would be appreciated, thanks!
 [/INST] Hints:
Notice that  $\sqrt{2x^2-2x+1}=\sqrt{x^2+(x-1)^2} \ge \sqrt{(\frac{x+1-x}{2})^2}=\frac12$ and 
$\frac{1}{x+\frac1x}\le \frac{1}{2x\frac1x}=\frac12$
</s>