2. The monotonic increasing interval of the function $y=2\left(\log _{\frac{1}{2}} x\right)^{2}-2 \log _{\frac{1}{2}} x+1$ is $(）$,
A. $\left[\frac{\sqrt{2}}{2},+\infty\right)$
B. $\left(-\infty, \frac{\sqrt{2}}{2}\right]$
C. $\left[\frac{1}{4},+\infty\right)$
D. $\left(-\infty, \frac{1}{4}\right]$
2. A $\log _{\frac{1}{2}} x$ is monotonically decreasing on $x \in\left[\frac{\sqrt{2}}{2},+\infty\right)$, and its range is $\left(-\infty, \frac{1}{2}\right]$. The function $y=2 x^{2}-2 x+1$ is monotonically increasing on $\left(-\infty, \frac{1}{2}\right]$, so the function $y=2\left(\log \frac{1}{2} x\right)^{2}-2 \log _{\frac{1}{2}} x+1$ is monotonically increasing on $\left[\frac{\sqrt{2}}{2},+\infty\right)$. Therefore, the correct choice is A.