Given ROC curve (http://en.wikipedia.org/wiki/Receiver_operating_characteristic), how do you read the maximum F1? (if it is possible)
The coordinates of points on the ROC curve are true positive rate (TPR) and false positive rate (TPR). These are given by: $$ TPR=TP/(TP+FN)$$ and $$ FPR=TP/(TP+FP)$$ The F1 score is given by: $$F_1=2*precision*recall/(precision + recall)$$ where $recall=TPR$ and $precision=1-FPR$.
So in order to calculate the maximum $F_1$ score, you have to calculate it at every point on the curve, where you take the $TPR$ value for $recall$ and the $FPR$ value for $1-precision$.