3. Let the function $f(x)=M \sin (\omega x+\varphi)(M \neq 0$, $\left.\omega>0,-\frac{\pi}{2}<\varphi<\frac{\pi}{2}\right)$ have a graph that is symmetric about the line $x=\frac{2 \pi}{3}$, and its period is $\pi$. Then ( ).
(A) The graph of $f(x)$ passes through the point $\left(0, \frac{1}{2}\right)$
(B) $f(x)$ is a decreasing function in the interval $\left[\frac{\pi}{12}, \frac{2 \pi}{3}\right]$
(C) A center of symmetry for $f(x)$ is $\left(\frac{5 \pi}{12}, 0\right)$
(D) The maximum value of $f(x)$ is $M$
3. C.

From the problem, we get $\omega=2, \varphi=\frac{\pi}{6}$. Therefore, $f(x)=M \sin \left(2 x+\frac{\pi}{6}\right)$. At this point, $f\left(\frac{5 \pi}{12}\right)=0$ holds.