The existence of prisons is a testament to this failure of humanity's ability to self-correct. They're a necessary evil, but they should never, ever be mistaken for a positive good, and alternatives should be sought whenever possible.
And yet you say concerning the death penalty:
Never in any circumstance.
For truly incorrigible criminals, the death penalty is theoretically a potential solution, tho in practice, one that is unlikely to be widely applied.
There is no such thing as a truly incorrigible criminal. Only a lack of time, will, and ability to make people see the error of their ways.
It's not societys job to rehabilitate every criminal. Youve only decided to bestow this responsibility on everyone apparently. We all only have one life to live. Some people choose to use their lives to harm others, and it's our job to make sure they can't ever harm anyone else again. Thats what prisons are for. We lock up the violent people that way us and our children can live our lives without fear.