Which feature allows some Macs to perform backups and check for emails while in sleep mode?

The feature that allows some Macs to perform backups and check for emails while in sleep mode is Power Nap. This functionality is designed to enhance the usability of Mac computers by enabling them to perform important tasks even when the system is in a low-power state.

When Power Nap is enabled, the Mac can periodically wake up to check for new emails, perform Time Machine backups, fetch updates from the App Store, and sync iCloud data. This capability helps users stay up to date without having to fully wake the machine, striking a balance between functionality and energy savings. The feature is particularly useful for maintaining operational continuity without the need for constant user interaction.

The other options do not provide this capability. Sleep is a state that saves energy but does not perform background tasks. Shut Down completely powers off the machine, disabling all processes and preventing any background operations from taking place. Safe Sleep, on the other hand, saves the current session to the hard drive before entering sleep mode but does not have the same background functionality as Power Nap. Thus, Power Nap stands out as the correct choice for enabling these specific tasks in sleep mode.