My Macbook Pro was purchased late 2013. I now have OS X 10.9.5 (13F34). All of my Apple drop down menu items have become dysfunctional. I consistently cannot shutdown or reboot.
What can I do?
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It sounds like your Mavericks installation is messed up.
If you still have the original Mavericks Install package you can check it with Terminal.
(with space) and now drag the Mavericks installer to the terminal window.
The result should be 7c2152a1afac38d2d06651add2d24968
I would recommend following steps:
Try to restore it using the cmd-r.
Following this, try to log in as different user to eliminate user profile problems.
As next step start in Safe mode to eliminate some of the 3d party apps problems.
Resetting the PRAM/NVRAN (EFI) and the SMC would be good as well.
Finally run the Apple Hardware test to check for hardware problems.