1. 1 Example 4 A road is to be constructed form city P to city Q as in the diagram below. The first part of this road PC lies along an existing road which costs $200,000 per km to renovate. The second part of this road CQ is new and costs $400,000 per km to construct. Where should C be chosen to minimize the cost of constructing this road? Solution In the above picture we denote the distance from P to C by x, measured in km. The right triangle has sides of length 5-x and 3. By the Pythagorean Theorem the length of its hypotenuse CQ is Let K denote the cost of constructing the road from P to Q. Then K is the sum of the cost of renovating the road PC plus the cost of constructing the road CQ: with domain [0,5]. The problem is to find the value of x which minimizes K.

2. 2 To find the critical points, we set the derivative of K equal to zero: since is not in the domain [0,5] of K. To find the minimum value of K we compare the values of K at the critical point and at the endpoints of the domain [0,5] of K:

3. 3 Hence K has its minimum value at