The intrinsic randomness (IR) problem is considered for general setting. In the literature, the optimum IR rate with respect to the variational distance has been characterized in two ways. One is based on the information spectrum quantity and the other is based on the smooth Rényi entropy. Recently, Nomura has revealed the optimum IR rate with respect to f-divergences, which includes the variational distance, the Kullback-Leibler (KL) divergence and so on, by using the informational spectrum quantity. In this paper, we try to characterize the optimum IR rate with respect to a subclass of f-divergences by using the smooth Min entropy. The subclass of f-divergences considered in this paper includes typical distance measures such as the total variational distance, the KL divergence, the Hellinger distance and so on.