Do the following sentences work with or without the prepositions?
This is a good place to stay (at).
This is a good place to live (in).
Do the following sentences work with or without the prepositions?
This is a good place to stay (at).
This is a good place to live (in).
I believe the prepositions would only be required when dealing with a specific noun.
For example: This is a good HOUSE to stay (in/at(in sounds more natural to me))
Or: This is a good HOUSE to live in.
Your two sentences are proper, no prepositions required.
They work both with and without the prepositions.
You'll find many articles with a search of "end sentence with preposition" on how teachers used to say this was wrong, but that the "old rule" of never ending a sentence with a preposition is a Latin grammar concept attempted to be forced into English's Germanic grammar roots, and therefore it's not really a big deal.
It helps clarify things as well. For example, This is a good place to live - we're not sure if you mean the neighboorhood, country, house, or what type of place, but by saying This is a good place to live in, it's clearer you probably mean a house or at least something with a roof.