The situation that is presented in the question is a good example of a situational irony.
This is a rhetorical devise that is used to show the difference that exists between what exists and what is expected from the actual reality of events.
In the question, her mother told her that so long as what she was writing was not about the family. But ironically, her daughter ended up writing the family biography.
Read more on irony here: https://brainly.com/question/11821145