Doctor dodge needs to see 65 patients next week; she works Monday through Saturday.
A- give a lower bound to the number of patients she will see on the busiest day next week.
B- give an upper bound to the number of patients she will see on the least busy day next week.
I'm really lost on this question, because I've been thinking it's only giving us how many days she's working but they don't say which busy day is it, it could be Monday-Saturday any day.
For $A$, notice that $65/6=10,8333\dots$ If our doctor saw $10$ or less patients each day, then she wouldn't be able to see all $65$ patiens. Hence, there must be a day she sees at least $11$ of them.
Now, if she sees $10$ patients on Saturday and $11$ on each other day, she will see all $65$ of them. This shows that it's possible for $11$ to be the number of patients seen on a busiest day of next week. The answer to $A$ is hence $11$.
A similar reasoning shows that the answer to $B$ is $10$. If it were higher, then she'd need to see at least $11$ patients each day... but then the total would be at least $66$, which is higher than $65$.