Let $ f: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \to \mathbb{R} $ be given by $ f(x) = \frac{\pi}{2} + x - \tan^{-1}(x) $. Consider the following statements:
$P:$ $ |f(x) - f(y)| < |x - y| $ for all $ x, y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) $
.
$Q:$ $ f $ has a fixed point.
Then the correct option is:
(A) both P and Q are TRUE
(B) P is TRUE and Q is FALSE
(C) P is FALSE and Q is TRUE
(D) both P and Q are FALSE
