Also, on one's good behavior. Very polite, as in Mother told the children to be on their best behavior during Grandma's visit, or The whole staff were on their good behavior while the client inspected the premises. [Late 1600s]
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.