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The Rice probability density function has received considerable attention for its various important technical and scientific applications. One of the more attractive techniques for extracting the distribution parameters, and possibly the most frequently applied, relies on the maximization of the likelihood function for a given set of experimentally determined samples, and many applications are documented in the literature. This paper offers a mathematical analysis which demonstrates that, subject to conditions universally verified in physical systems, an absolute maximum exists, and it is the unique point internal to the domain of existence which zeroes the gradient of the likelihood function. In all previous results, the presence of additional maxima, which are possibly larger than the one that had numerically been found, could not be excluded. We can incidentally state that this paper demonstrates that all previous results based on numerically finding a maximum indeed corresponded to absolute maxima. The mathematical derivations offered here are also suggestive of actions capable of improving the insight into the maximum-likelihood technique and its numerical implementation.