I have the Canon 70-200mm f/2.8 L and the Canon 400mm f/5.6 L. I also have the Canon 1.4x II extender, and was able to borrow a Canon 2x II extender from a friend for a couple days about a week ago. So naturally I decided to test these two lens with different combination of the extenders to see how they preformed. I have been planing to write up a review with my findings, but I'm a college student so I have been very busy.
Seeing how you need help decided what to do I figured I can show you some of the results from the test. Now keep in mind that I don't test lens all the time and this was my first time doing something like this. But I did do the test multiple times to ensure that I got a good sample.
Here is the uncropped comparison of the different focal lengths all with the 70-200 f/2.8 non IS. 200, 280(1.4x), 400(2x)
Now here are the 100% crops all of which were stopped down 1 stop from wide open.
Now at 400 the image quality is fair, some samples at 400 were slightly better, while some were slightly worse. I did let the camera use auto focus. But in general the results show a loss of image quality and that is to be expected.
With that said I did go out and take field shots and here is a comparison between the 70-200 with 2x extender and my 400, both taken at f/8
Now in the field I find it hard to see a huge difference between the two.
The person that I borrowed the 2x extender from uses it on her 70-200 f/2.8 II, the same lens you got and she seems to be happy with it. Any time we go take pictures she puts it on and takes nice pictures. So hope this helps, I'll try and check back to answer any questions you might have.