You would definitely be having a hard time coping with a child who is perfect to the outside world and a difficult child at home. It is not easy to pinpoint the exact reasons for this difference in her behaviour, but since the problem started 3 years ago, it may be linked to
a) birth of the youngest sibling or
b) pre-adolescence.
Children these days are going through adolescent traumas earlier than before. A confusion in self-identity and a rebellious attitude at home are some indicators. My suggestion would be to get a complete psychological assessment done. She needs to be listened to very carefully to understand what themes underlie her stubbornness and defiance. The best alternative in dealing with her is to try to ignore her undesirable tantrums and negative comments. She is probably using them as attention-getting devices. On the other hand, try to give her adequate warmth and attention when she appears to be calmer and in a good mood.
Finally do remember that children go through phases and hopefully she will come out of it soon. Try to be patient and serve as good role models yourselves.