I think it's wrong to "give up" on anyone regardless of kid/adult bio/adopted. Unfortunately the children's mental health clinic thru my county often encourages parents to sign court paperwork declaring their kid incorrigible; claiming it will help their child get services paid for by the state. When Angel was 12yo they were trying to get me to sign off on her, it wasn't to get her help it was to get her off their caseload because she was a PITA.
It doesn't get the kid services, in reality it terminates the parental rights and often these kids sit in Juvi until they are 18yo. It happened to my neighbor all they wanted to do was scare the kid - 8 months and several thousand dollars in legal fees and still couldn't get back custody of kid. The kid ended up staging a suicide attempt in juvi to get himself transferred to a psychiatric hospital and that is the route they had to take to get their kid back.
Another family having exhausted all services available fell into that trap and signed off on adopted 13yo (as advised by psychiatrist at clinic) here you can do that with a bio child but not an adopted one. The state kept the child in the home and charged every family member except her with abuse charges. Long story short ... girl didn't get into the Residential Treatment Center (RTC) she needed desperately, father died of heart attack during one of girls blow outs, mother $15,000 legal fees & 5 years probation (with-no contact anyone under 16yo) and her 15yo brother another $40,000 legal fees and sat in the state maximum juvi system until he was 21yo (only contact with mother was saw once across courtroom - they wouldn't allow her to see her son the entire time incarcerated.) Now the mother & son live together, the girl is a prostitute and drug addict who has 3 kids in the state foster care system - I sure wish someone had gotten her the mental health treatment she needed when their was still hope for her. Good chance the 3 drug addicted babies that were taken away from her would agree.
I've lived with an out of control kid beating the **** out of me every day and endangering the lives of everyone under the roof with her. I was lucky that Angel just cut her sisters hair with steak knife and not her throat; lucky didn't break my neck when swung full body weight from my hair. I believe only someone who has actually lived in that situation could even begin to understand how a parent could be pushed into dumping a kid. Looking back now I realize by not dumping Angel back then that I had gambled with my other 2 children's lives; glad I won that time but explains why I don't gamble (I can't afford to lose).
Sorry for this being so long, you all have a peaceful weekend
Nancy