Regarding Background activity manager on 7.1.2 Update.
You can choose which applications you want to run in the background. Like running the Mail application to continuously run in background and notify you as soon as u get a mail. Else you will get the mail / notification updates only after opening the application manually.
It surely is a good way to stop applications sucking battery and checking for mail/ updates if you have limited mobile data.
Also, Check Settings > Battery > Battery Activity Manager > 'Application' should be added in Whitelist (So it can run in the Background).
PS :I noticed that after each application is updated, It is automatically added to Whitelist !!