What is the minimum population required to keep a language alive?
Theoretically, there isn't a minimum population for a language to survive, because a speech community can theoretically be of any size. Even if the speech community dwindles to one person, that last person isn't going to stop transmitting his language if he isn't exposed to any other languages.
However, this isn't a practical case, because the last speakers of a language are going to be exposed to other languages. Speakers of any language, no matter what number of people speak it, will tend to gravitate towards more prestigious languages. This can happen to a language with any number of speakers; for example, speakers of Chinese who immigrate to the United States will tend to learn English, and eventually may stop transmitting Chinese to the next generation, because English has more prestige than Chinese in the United States.
But for a language to become extinct, all speakers would have to give up the language. This would usually happen in the case of minority languages whose speakers are all exposed to a more prestigious language (e.g. aboriginal languages spoken in a country with a different predominant language).
As for the actual number of speakers necessary in practice for a language to survive, I don't know what, if any, empirical data there is. But since you are looking specifically for the minimum, in optimal conditions there is none.
Language transmission can occur in different ways. Generally, a language is learned inside a group social. So, theoretically, without considering external considerations (peer pressure, motivation, ...), it can be two persons at least a learner and a teacher without having necessarily a familial relationship; or two genitors who give birth to children and transmit their language. If this condition is met at each generation, so it can survive.