[See also comments about the Gospels in general, and the book of John in particular, under the reference Matthew 6:33 – in the “Happy Conclusions” subsection]
Jesus frequently had to respond to people who challenged his authority as a teacher and guide. This passage is one such response. My guess is that the reason the passages put under the heading of “When Doubtful”, is that the people Jesus was talking to at this point were in fact doubtful themselves about the quality of his teaching, and it’s value in the community of his day.
The argument Jesus uses is a very interesting one – it’s the argument from practice. The Greeks were very much into what we would call orthodoxy – that is right belief, or correct belief. The Hebrews were into what is called orthopraxy – right or correct action (practice). What Jesus says is that anybody who’s into trying to move their life into a better quality practice/action, will know intuitively whether his teaching comes from God or from ordinary people – because they will recognize right away the spiritual quality of the practice that he encourages people to take up as a way of life.
The quality of the lifestyle that he is recommending is a quality which gives glory to God not glory to himself as a speaker and teacher. If there is a ring of truth (not falsehood) about this non-hypocritical lifestyle he is recommending, then it is pretty evident that the source of the teaching is God.
In other words, he is saying, if you doubt my words, maybe ask yourself if you’re really into upgrading quality lifestyle/practice – because in the end actions speak louder than words. Doubt and belief are all very well and good, but in the end the proof of the pudding is in the eating. Look closely at the lifestyle of the people teaching, and of the lifestyle which is being promoted – that’s where you check out the quality, and that should be the focus of our doubt and belief.