How to prove a function is pseudorandom function?
As it is, there is not enough information, in particular on functions A and B, to answer. However, here are elements that may help:


  
  Is the above function using A and B a pseudorandom function as it is using LFSR to produce cipher text?
  


As mentioned in the comments, even if the LFSR does outputs completely random numbers (which I doubt), there is no guarantee that F is pseudorandom. As far as we know, A and B could be deterministic functions, always outputting, say 42.


  
  If the above function is pseudorandom then how to prove it mathematically?
  


You would have to use a proof by reduction, such as one depicted in this video tutoral. That is, you do not prove directly that your function F is pseudo-random. Rather, you would prove, in your case, that IF the LFSR function (or function B maybe) is pseudo-random, THEN F is pseudo-random.


  
  Can any function which uses LFSR as the random number generator be considered CPA secure?
  


IND-CPA is a property for encryption. There is some relation between encryption and pseudo-randomness, but not as you seem to be saying. Generally, an encryption scheme need to use a good pseudo-random number generator (or a truly random number generator) when used in practice, but that does not take part in the cryptographic proof of IND-CPA. Thus, short answer here would be: no, even if LFSR is somehow truly random. You need other assumptions, and to study a particular, fully defined encryption scheme.

Hope this helps. It may be possible to answer more clearly if the functions $A$ and $B$ were defined.