Two towns at each end of a road have populations of 12,000 and 2,000
respectively. A library is be constructed along the road to serve both
towns. Where should the library be located to minimize the average
access distance?

That's easy. The library should be built at the end of the road with
the larger town. Here is the math, no calculus is needed:

We assume that the probability of anyone visiting is independent of
the town and independent of how far away the library is. The average
distance traveled is:

D = (12000*x + 2000*(L-x))/14000,

where x is the distance from the larger town to the library and L-x is
the distance from the smaller town. L is the distance between towns.
This can be simplified to:

D = (5*x + L)/7

This is clearly smallest if we set x=0. In general, you should always
put the library adjacent to the larger town. If the towns are the same
size, you can put the library anywhere on the road.

Kottekoe is correct and beat me to post an anwer.  The library should
indeed be built in the larger town.  And as he observed, no calculus
is needed.

Can't fault the maths....but leads to an interesting social issue.

Imagine the towns are 10,000 people and 10,001 people.

The minimum average distance will still be to build it in the town
with 10,001 people...but socially everyone will expect it to be just
about half-way.

John

kottekoe is right.

This is a calculus problem in its infancy: but to answer it as is -
just ask yourself: is it better to have 12000 people drive x miles or
2000 people drive x miles?

Have a good day.