When I use Contour > Nodes at Extremes it doesn't work hence FontAudit suggests to add a node at the extremum. Why?
Nodes at Extremes will not add extreme points if doing so will damage the contour.
Sometimes Y-delta between the needed extreme point and the nearest on-curve point is not enough to define the "short" curve.
Since Nodes at Extremes is a "batch-processing" action, it is very conservative in cases where adding an extreme point may create kinks in integer curves. In this case, the curve should be modified with slide, instead of adding the extreme point. Alt-double click to the handle will do exactly that.
Doing that automatically is not a good approach as it may modify the curve.