Carrier APU

For you 110v (which is powered from the generator, it's not an inverter), that plug will have a Ground Fault Circuit Interrupter (GFCI) plug... between the two 110 plugs, you'll see two buttons... one will say "Test", and the other will say "Reset". Push the "reset" button first to see if you simply tripped the GFCI. If that doesn't do it, pull off the cover at the bottom of the unit... get a flashlight, look for a red button.. push it in, especially if you don't have heat... there's a capacitor down there.. I'll try to get a photo this weekend... that often blows out.
Check your belts, as well.. the APU should be serviced every 1000 hours, and the OEM recommends replacing the belts with each service. Older models use two black belts... one for the generator, one for the water pump, fan, and alternator... newer ones use a different style generator belt, which is a red linked belt. You can add tension to the fan belt (on both models) with a 12mm wrench, 12mm socket, ratchet, and extension of a decent length by adjusting the alternator to add tension... on the older models, the tensioner for the generator belt is accessible through the bottom of the housing... how I like to do it.. I take a 13mm socket (and my air ratchet, but a manual one works, as well), take off the entire bottom section of the housing... some people like to remove the air filter housing and pull that back... you'll need a 3/4 wrench, and you'll loose the lower two nuts, move them down, then tighten the upper two... on the newer models, it's kind of a pain in the nuts.. there's a special tool you use to remove that belt from the APU, and you have to take both sections of the housing off... then you remove the belt, and, to tighten it, you remove a link and reassemble the belt.
If the belts look kosher, VERY CAREFULLY[/i] look on the bottom of the APU towards the front of the vehicle, you'll see an output from the generator with two red caps.. remove those caps, and you can test for voltage there with a DMM.. you should be getting 120 - 140 volts there. Again, be extremely careful, unless being a crispy critter appeals to you.
After that, the next area of focus moves to the CECU, which is that box under the bunk... first, clean the CECU filter (because nobody else will). Ensure all the plugs are secure, run the unit.. there'll be lights which will indicate when the 110v outlets are operational, there'll be lights for AC low, AC high, heat low, heat high.... if things are asunder there, that goes well beyond operator level, and you need to bring it in at that point.