Okay Gents, I like the models, but gonna try to be more specific.
You takeoff and lose an engine you take the radios and ask your monitoring to troubleshoot. While he/she starts doing that you go ahead and tell ATC that you wanna stay around the airport for a subsequent return. Easy.
Now lets say you takeoff and get a caution thats not so cut and dry. You might end up coming back to the airport, you might just continue rendering it an indiction failure and monitor it the whole way. Do you right away declare that you'd like to stay in the vicinity of the departure airport? Do you tell them you want a slow speed while troubleshooting?
How do you decide when you start talking on the radio vs troublshooting all the while keeping in mind you don't want to get too far out from a good airport with good runways.
Thank you for all your replies!
Beamer: I'm afraid i'm part of those 'non civilized' ones ;-) [all in good spirits here!]