Let $Q = (0,3)$ and $R= (9,5)$ be given points in the plane. We want to find the point $P=(x,0)$ on the $x$-axis such that the sum of distances $PQ+PR$ 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 $x$:
$f(x)=$
over the closed interval $[a,b]$ where $a=$ and $b=$.
We find that $f(x)$ has only one critical number in the interval at $x=$
where $f(x)$ has value
Since this is smaller than the values of $f(x)$ at the two endpoints, we conclude that this is the minimal sum of distances.