Yes I believe that cp's exist at the endpoints of a closed interval because by definition they are undefined
I don't know what a cusp is, but for my first answer I am not 100% sure as cp's are supposed to be found on the open interval (a,b). I think you find them on the open because on a closed they are cp's because the instaneous slope is undefined, but this does not paint an accurate picture as the function could be continous for all reals and have a defined slope. I don't see the problem though as you test cp's AND endpoints for max/mins. That might be why it's on the open because it is already fact that the endpoints ARE undefined in slope so...
I looked up cusps and got this definition:
Mathematics. A point at which a curve crosses itself and at which the two tangents to the curve coincide.
I have no clue to be honest, I haven't gotten that far in math. I would take it as a cp but don't come close to trusting me on this one sorry. I can't give a reason, but if that is your endpoint then yes for the above reasons