Apathy is characterized by symptoms such as reduced emotional response, lack of motivation, and limited social interaction. Current methods for apathy diagnosis require the pa-tient's presence in a clinic and time consuming clinical interviews, which are costly and inconvenient for both patients and clinical staff, hindering among others large-scale diagnostics. In this work we propose a multi-task learning (MTL) framework for apathy classification based on facial analysis, entailing both emotion and facial movements. In addition, it leverages information from other auxiliary tasks (i.e., clinical scores), which might be closely or distantly related to the main task of apathy classification. Our proposed MTL approach (termed MTL+) improves apathy classification by jointly learning model weights and the relatedness of the auxiliary tasks to the main task in an iterative manner. Our results on 90 video sequences acquired from 45 subjects obtained an apathy classification accuracy of up to 80%, using the concatenated emotion and motion features. Our results further demonstrate the improved performance of MTL+ over MTL.