Prove that
$$\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}$$for $0 < x < 1.$
I've tried to find some intermediate expression where $\sqrt{\frac{2x^2 - 2x + 1}{2}} \ge \text{something} \ge \frac{1}{x + \frac{1}{x}}$, but I can't figure out how to get it working. Squaring both sides will only create a huge mess.
Any help would be greatly appreciated. Thanks!