On the unimodality of the likelihood for the Cauchy distribution

Although the likelihood for the median of a Cauchy distribution with known scale is often multimodal, the joint likelihood for both location and scale parameters has exactly one point of maximum and at most one stationary point. The maximized likelihood function is therefore unimodal. It is also suggested that the likelihood for the location, when evaluated at the maximum likelihood estimate of the scale, is unimodal considerably more often than might have been anticipated from the single-parameter case.