Nowadays, a large amount of services are deployed on the edge of the network from the cloud since processing data at the edge can reduce response time and lower bandwidth cost for applications such as healthcare in smart homes. Resource management is very important in the edge computing since it is able to increase the system efficiency and improve the quality of service. A common approach for resource management in edge computing is to assign tasks to the remote cloud or edge devices just according to several factors such as energy, bandwidth consumption, and latency. However, the approach is insufficiently efficient and falls short in meeting the requirements of handling health emergency when being applied in smart homes for healthcare. Possible health emergency needs immediate attention and different health tasks have different priorities to be processed. In this paper, we propose a task scheduling approach called HealthEdge that sets different processing priorities for different tasks based on the collected data on human health status and determines whether a task should run in a local device or a remote cloud in order to reduce its total processing time as much as possible. Based on a real trace from five patients, we conduct a trace-driven experiment to evaluate the performance of HealthEdge in comparison with other methods. The results show that HealthEdge can optimally assign tasks between the network edge and cloud, which can reduce the task processing time, reduce bandwidth consumption and increase local edge workstation utilization.