1 Timothy 5:1-2

1Do not sharply rebuke an older man, but rather appeal to him as a father, to the younger men as brothers, 2the older women as mothers, and the younger women as sisters, in all purity.

Looking back at 1 Timothy 1:3-4, we see that Timothy is going to have to instruct some men not to teach "strange doctrines" and engage in conversations that aren't really helping anyone. From these passages 2 things stand out:
1. Timothy has authority as being appointed to his position by Paul and church elders.
2. Timothy is young (4:12).

Timothy being young and most likely single means he had 2 cultural strikes against him: In Asian cultures, like those found in Asia Minor, real adulthood begins when you're married. Not only that, but authority is always given to the oldest male.

Yet, he had God-given authority, and the older men in the church would have to submit to him if they wanted to be obedient.

I can't imagine having to correct an older man, even by "appeal"ing to him as a father. Have I ever even "corrected" my own father? I was raised in obedience to his position of authority as head of our household.
Timothy, as a very young and single man, is now the authority of the church in Ephesus. Tough job, I'm sure, which is why Paul continually encourages him.

I would find it even harder to talk to a woman about something she's doing wrong. Doing this tactfully is a tough task, and I'd probably send my wife (someone who understands her own gender better than I) to say "You know, what you're doing isn't really benefiting the body..." Timothy didn't have that option.

It's like Paul is saying "be tactful, be respectful, and don't be harsh." This would mean Timothy needed to have patience and firm reliance on the Spirit as well as a boldness to carry out the task he was given.