One train route follows the curve
y = 2x2 − 4x + 6,
and the other follows y = x2. Assume that distance along both axes is measured in miles. The railroad wants to construct a north-south maintenance road PQ between the two routes. Where should the road be located so that it is as short as possible?

Question

One train route follows the curve
y = 2x2 − 4x + 6,
 and the other follows y = x2. Assume that distance along both axes is measured in miles. The railroad wants to construct a north-south maintenance road PQ between the two routes. Where should the road be located so that it is as short as possible?

tkhunny · Answer

It is very fortunate that this problem states "north-south".  That is, if we want the problem to be easier.  On the other hand, if it didn't say that, it would be a MUCH more intereting problem.

Just minimize the vertical distance: $M(x) = (2x^{2}-4x+6)-(x^{2})$