Find a point on the graph of and a point on the graph of such that the distance between them is as small as possible.

To solve this problem, we let be the coordinates of the point . Then we need to minimize the following function of and :

After we eliminate from the above, we reduce to minimizing the following function of alone:

To find the minimum value of we need to check the value at the following three points (in increasing order). (You will need to use a numerical method, like Newton-Raphson to find one of these points.)

We conclude that the minimum value of occurs at

Thus a solution to our original question is


You can earn partial credit on this problem.