It is Caruana's turn to move. This should have been a draw, and thus the match would have been so different. So what is the drawing plan for Black?
This is where it started to go off the rails for Caruana, but was it really the losing move? This ending is well beyond my ability to fully analyze. According to the Stockfish 8 analysis engine on Chessbomb.com (where I watched the games yesterday), it does suggest 34......c3 as a clear draw, and when you play through the main line in that variation, it is a way for black to force the win of the white g-pawn in exchange for the passed black c-pawn- then the white e and f pawns also fall leading a fairly standard draw.
However, isn't the real losing mistake the one at move 37 for black? It is the move that allowed Carlsen to force Caruana to win the f-pawn with the king rather than with the rook- this is what allowed Carlsen to win both the g and h pawns creating the connected passers on the king side.