Brain Teaser - Minimum distance between two curves

[CR Haeger]

Finding the point on a curve closest to a fixed point off the curve seems to be fairly straightforward and often can be solved by hand. Finding the points on each of two curves which are closest seems to be a bit more involved.

Problem: What points on each of the two curves f(x) = x^2+1 and g(x) = √(x) are closest to each other?

   

I worked this one mainly by hand, then used the HP Prime CAS when things got messy (and they did for me). Can you solve this by hand, by CAS, programming or other methods?