In a recent contribution, Horváth and Trapani (2022) develop a family of CUSUM-based changepoint tests in the context of a Random Coefficient Autoregressive model of order 1. In this paper, we complement the results in that contribution by studying the (maximally selected) Likelihood Ratio statistic. We show that this has power versus breaks occurring even as close as O(log log N) periods from the beginning/end of sample; moreover, the use of Quasi Maximum Likelihood based estimates yields better power properties, with the added bonus of being nuisance-free. Our test statistic has the same distribution - of the Darling-Erdős type - irrespective of whether the data are stationary or not, and can therefore be applied with no prior knowledge on this. Our simulations show that our test has very good power and, when applying a suitable correction to the asymptotic critical values, the correct size. We illustrate the usefulness and generality of our approach through applications to economic and epidemiological time series.