If it is a Mac, that is an option in energy settings in system preferences.
If it is a PC, different manufacturers and models may (or may not) offer this option. It is most often used in unattended (server type) operations. You might find the answer in your manuafacturers web site.
Like Whiz said, It is, in most systems, a bios setting. In addition to the keys he listed, I have seen the "Esc" key, the "Del" key and even an "F2" key being the entry point into the BIOS. When you come up from a cold start or a restart, look closely at your screen. The correct key for your system should be shown for a second or two. Most likely it will be telling you what key to hit to enter "Setup" mode. It will not say "BIOS" unless you have some other BIOS that I have not run across.
Just be sure to only change the Power setting for power failure restart. There are probably more than a hundred different BIOS's in use and every manufacturer may change the exact name and how you get to it. Just read many times BEFORE changing once. Do not make any other changes unless you know exactly what you are changing and why. Once you make THE change you will have to save the new setting in BIOS and then restart.
I have seen a one or two systems where the option wasn't offered but those, as near as I recall, were on laptop systems and it was only one or two times. I would think that in a laptop it would only apply if both the AC adapter or docking station failed and the internal battery was totally dead or removed. There may have been more but I wasn't really looking for or paying attention to that setting.
As it was easier to access the bios setup, I tried that first. The AC Power Recovery option was set to "last". I changed it to "off", saved the change and rebooted. It did not work. The computer turned itself on when I restored power. During the boot process, there is a screen indicating the need to enter setup to change the clock setting, etc. This screen also indicated: "Alert: system battery voltage low". Therefore, it seems to me that Pyrdek's original concern may the correct one.
Is this something that is user serviceable, (I would classify myself as reasonably competent), or should it really be done by a fully qualified repair tech? Thank you both for all your help.