Detecting features in XY Plots using CNNs
I am a bit surprised that fully connected neural networks are unable to learn this. It might be due to one or more of the many possible problems like few data or overparameterisation.

Keeping this aside, your problem seems to be solvable by tree classifiers like CART, Random Forest, Gradient Boosting Machine or XGBoost. 

The reason I think tree methods can work is that the problem can be solved using a decision tree. To detect the kinks one needs to find whether the datapoint is below the red line within some specific range of X and Y. Once the kink is detected it is easier to predict TRUE or FALSE as you mentioned.