Let \( f(x)=x \cos ^{-1}(-\sin |x|), x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), then which of the following is …

\( f^{\prime}(0)=-\frac{\pi}{2} \)

\( f^{\prime} \) is decreasing in \( \left(-\frac{\pi}{2}, 0\right) \) and increasing in \( \left(0, \frac{\pi}{2}\right) \)

\( f \) is not differentiable at \( x=0 \)

\( f^{\prime} \) is increasing in \( \left(-\frac{\pi}{2}, 0\right) \) and decreasing in \( \left(0, \frac{\pi}{2}\right) \)