The task search page is designed to help you quickly find specific incidents based on the status of their tasks. You can search by Deadline Date to find tasks based on the date they are due, or you can search by Completed Date to find tasks based on the date users marked them as complete (if the date exists). You can also filter the status by Complete or Incomplete using the radio buttons on the right side of the page. The criteria you provide will search for any task tied to an incident, so if you search for tasks with a Completed Date after January 1 but filter by Incomplete Tasks, you'll find incidents where some of the tasks have been completed but have at least one task incomplete.
The dates you provide are only inclusive when you use the Between criteria. In other words, if you search for all incidents with tasks that have deadlines between January 1 and January 7, the results will include those dates. However, if you search for tasks with deadlines After January 1, your results will start at January 2. If you search for tasks with deadlines Before January 1, your results will start at December 31 and count backwards.