2010 AMC 12A Problems/Problem 22
Problem 22
What is the minimum value of 
?
Solution
If we graph each term separately, we will notice that all of the zeros occur at 
, where 
is any integer from 
to 
, inclusive.
The minimum value occurs where the sum of the slopes is at a minimum, since it is easy to see that the value will be increasing on either side. That means the minimum must happen at some .
The sum of the slope at 
is
Now we want to minimize 
. The zeros occur at 
and 
, which means the slope is 
where 
.
We can now verify that both 
and 
yield 
.
