Let $\displaystyle {Q}={\left({0},{5}\right)}$ and $\displaystyle {R}={\left({6},{10}\right)}$ be given points in the plane. We want to find the point $\displaystyle {P}={\left({x},{0}\right)}$ on the $\displaystyle {x}$-axis such that the sum of distances $\displaystyle {P}{Q}+{P}{R}$ 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 $\displaystyle {x}$:
$\displaystyle {f{{\left({x}\right)}}}=$Preview Question 6 Part 1 of 5  
over the closed interval $\displaystyle {\left[{a},{b}\right]}$ where $\displaystyle {a}=$Preview Question 6 Part 2 of 5   and $\displaystyle {b}=$Preview Question 6 Part 3 of 5   .
We find that $\displaystyle {f{{\left({x}\right)}}}$ has only one critical number in the interval at $\displaystyle {x}=$Preview Question 6 Part 4 of 5  
where $\displaystyle {f{{\left({x}\right)}}}$ has value Preview Question 6 Part 5 of 5  
Since this is smaller than the values of $\displaystyle {f{{\left({x}\right)}}}$ at the two endpoints, we conclude that this is the minimal sum of distances.