Best answer by Chris Kelly
You have to understand that if the client cannot 'hear' the AP, then there's a really good chance that the AP won't be able to hear the client either.
So just because you turn the AP up to 11 doesn't mean that the AP will be able to hear the responses coming back from the client.
Ideally, you want your AP's power settings to match those of your most important, least-capable device. Otherwise, you start dealing with issues caused by asymmetric power levels.
For now though, you can try bumping up the power on the AP a little to see if it might help (it might possibly). But, in doing some, you're increasing the cell size around that AP which will then cause the other APs to shrink theirs or possibly change channels (SmartRF) which is disruptive to client associations.
Bottom line, the real solution is to place an AP closer to the dead spots.