Sometimes when I hear someone say 'try and shut the door' I visualise the mental exertion the person must be going through as they 'try and shut it'. When did we change our language from 'please shut or close the door' to suddenly being seen as a challenge?
Teachers and coaches will often use 'try' as a way of encouraging someone to push beyond a perceived barrier. "try and do it..."
Common ones heard include "Try harder"," Try and put more effort into it", "Try and behave" I believe using 'try' actually creates a mental barrier for the person in some situations will make it harder for the person as they put their energy into the 'trying' bit.
I believe it would be more effective to remove the 'try' and focus on what you are wanting, a ...success or a change of some kind.
If I was working on a particularly thorny problem, to have someone say to me "try harder to solve it" would not make an iota of difference to whether I was able to solve it or not. However, it might make me question whether the person thought I wasn't already working hard to solve the problem!
Telling someone to 'try and behave' could be more effective if you name the behaviours(s) you want rather than some vague concept. "I want you to sit quietly for 10 minutes" or "please close the door quietly" (this is particularly effective, if you have a door slammer child!) will achieve more success because it is naming the behaviour you want and is unambiguous.
As an aside, it struck me how confusing it must be for second language learners, when they hear a commentator at a rugby game state X has 'scored a try!'