We regularly transport goods from city $A$ to city $B$, which is $183 \mathrm{~km}$ away. City $A$ is $33 \mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per km on the river is half that on land. Where should we build the road to minimize the transportation cost?
Let $C$ be the point where the road reaches the river, and $D$ be the point by the river such that the segment $DA$ is perpendicular to the river. Then $BD=180 \text{~km}$. If we denote the distance $CD$ by $x$, then $AC=\sqrt{33^{2}+x^{2}}, BC=180-x$. Our task is to determine the minimum point of the function $x \mapsto \sqrt{33^{2}+x^{2}}+\frac{180-x}{2}$ on the interval [0; 180]. This is possible, for example, by differentiation, where $x=11 \sqrt{3}$ is the minimum point. (The derivative function changes sign around $11 \sqrt{3}$, which is why the function indeed has an extremum there. Many people forgot to check this.)