Let and be given points in the plane.
We want to find the point on the -axis such that the
sum of distances is as small as possible. (Before proceeding
with this problem, draw a picture!)
To solve this problem, we need to minimize the following function of
:
over the closed interval where
and .
We find that has only one critical number in the interval
at
where has value
Since this is smaller than the values of at the two endpoints,
we conclude that this is the minimal sum of distances.