My iPhone 5 running on iOS 7.1.1 is doing the same thing since the update. I also noticed a few days ago that my buttons have become somewhat randomly unresponsive. Also, when it crashes, I am not able to power it back on. Screen goes black, spinning circle appears and then goes black and it's DONE!
It is like it is totally power depleted. Then, it will come back on randomly when I plug it into my computer via USB, or sometimes press and hold the power button, but neither consistently.
I have been able to back mine up as of today. I'm not familiar with the restore process. What will the restore do? Will it completely rephrase the phone and then restore it back to it's pre-restored condition ( hopefully minus the bug that's causing the crashes)?