I am using Illustrator CS6 and when I want to make a rectangle's corners round by using the direct selection tool and doing it manually, there is no option for that. So how to do so?
Live Corners and Live Shapes were new features added in CC so aren't available in CS6.
What you can do is use a "Round Corners" effect:
Effects → Stylize → Round Corners...
Since this functionality was not made available until later versions of illustrator, your only option is to do it through a plugin:
VectorScribe, from AstuteGraphics has not only live (rounded) corners, but TONS of functionality to edit paths. It is a must for all serious AI users.
XtreamPath is also a multi-function plugin that has rounded corners by dragging.
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1The reality is even with CC+, VectorScribe still handles this better than the built in Adobe functions. – Scott Dec 2 '16 at 11:05
I am using Illustrator CS5. I always do it like this. Then if I need to adjust further I use the Direct Selection tool and adjust it.
Live corners
are a CC feature. – Joonas Dec 2 '16 at 7:04