Add one rounded corner effect to a point of a segment or a shape as in indesign
I am unsure about the latest version(s), but I am afraid you are out of luck.
If you can use an actually rounded corner (two Anchor Points instead of one), regardless of version you may use the free Round Any Corner script available here:
You can use the Appearance panel and fills to fake the rounding of corners.
This method gives one a single or double rounded corner simply by clicking a Graphic Style. However, to change the color you have to alter 2 fills for each object in the Appearance Panel and to edit the rounding you may need to do a bit more tinkering with the appearance.
If you're interested, the AI CS6 (v16.0.0) file can be downloaded by clicking this text. Sorry about the overly ad-filled free file host, but I figure that's better than serving the files from my own server.
A workaround would be to select and cut the corner point, paste in front and use the Round corner effect. However to join the segment to the rest again, the appearance must be expanded which gives you the same result as if using the Round Any Corner script. The only advantage of the workaround is that you can keep the effect live for a time being before the final decision.
Honestly, the absolute best workaround is to purchase the VectorScribe plug in from www.astutegraphics.com. Then this entire issue is no longer an issue.
However, Astute Graphics hasn't released a CS6 compatible version of the plug in yet.