Challenge: Interesting Question (Part 2)

[enpassant]

use wolfram-alpha to minimise f(a,b)=sqrt((a-b)^2+(a^2+1-sqrt(b))^2)

alternative method:

let (a, a^2+1), (b, sqrt(b))

gradient=2a=1/2sqrt(b), .: a=1/4sqrt(b) and sqrt(b)=1/4a

grad of line joining the points=m=(a^2+1-sqrt(b))/(a-b)

2am=-1 and m/2sqrt(b)=-1

solve for a and b

a=0.331695, b=0.568071