You can use utmost to emphasize the importance or seriousness of something or to emphasize the way that it is done.
It is a matter of the utmost urgency to find out what has happened to these people.
当务之急是弄清楚这些人出了什么事。
Security matters are treated with the utmost seriousness.
安全问题受到极为严肃的对待。
If you say that you are doing your utmost to do something, you are emphasizing that you are trying as hard as you can to do it.
He would have done his utmost to help her.
他本来会尽最大努力帮助她。