This will be the last problem of the Fall Semester.  The Problem of the Week will resume with the beginning of Spring Semester classes at the end of January 2002. 
Happy Holidays!!

Start with the graph of y = x², at any point other than (0,0) draw the normal line to the graph; that is, the line which is perpendicular to the tangent line there.  This line will intersect the parabola at one other point creating a region similar to that sketched in gray on the left.  At which point on the parabola will the area of this region be the smallest?  (Before you begin, make sure you understand why there will indeed be a minimum value for this area!)

For the more inspired:  What is the corresponding minimizing point if the graph of y = x² is replaced by that of y = x²ⁿ for n any positive integer?  In particular, to which point does this minimizing point tend as n tends to infinity?