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

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

After we eliminate [math] from the above, we reduce to minimizing the following function of [math] alone:
[math]
To find the minimum value of [math] 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.)
[math]
[math]
[math]
We conclude that the minimum value of [math] occurs at
[math]
Thus a solution to our original question is
[math] [math] [math]
[math] [math] [math]