Consider the following three statements for the function \( f:(0,\infty)\to \mathbb{R} \) defined by \( f(x)=|\log_e x|-|x-1| \):
(\( \mathrm{I} \)) \( f \) is differentiable at all \( x>0 \).
(\( \mathrm{II} \)) \( f \) is increasing in \( (0,1) \).
(\( \mathrm{III} \)) \( f \) is decreasing in \( (1,\infty) \).
Then:
