Let
$$f(x) = \frac{\sin \pi x}{x^2}, x > 0.$$
Let $$x_1 < x_2 < x_3 < ... < x_n < ...$$ be all the points of local maximum off
and $$y_1 < y_2 < y_3 < ... < y_n < ...$$ be all the points of local minimum off.
Then which of the following options is/are correct?

A

$$x_1 < y_1$$

B

$$x_{n + 1} - x_n > 2$$ for every n

C

$$x_n \in \left(2n, 2n + \frac{1}{2}\right)$$ for every n

D

$$|x_n - y_n| > 1$$ for every n