Brain Teaser - Minimum distance between two curves

[Pekis]

Thanks for your refreshing brain teaser.

f(x)=x²+1 => f'(x)=2x
g(x)=sqrt(x) => g'(x)=1/(2sqrt(x))

The normal equation on f at x_f: Slope: -1/f'(x_f) Intercept: f(x_f)+x_f/f'(x_f)
-> y=-x/(2x_f))+x_f²+3/2

The normal equation on g at x_g: Slope: -1/g'(x_g) Intercept: g(x_g)+x_g/g'(x_g)
-> y=-2sqrt(x_g)x+(2x_g+1)sqrt(x_g)

Same normal => -1/(2x_f) must be equal to -2sqrt(x_g) and x_f²+3/2 must be equal (2x_g+1)sqrt(x_g)

Leads to
x_f=1/(4sqrt(x_g)) (or x_g=1/(16x_f²)
and then
4x_f⁵+6x_f³-x_f²-1/8=0

Only one real root: approx. 0.331695 for x_f => approx. 0.56807 for x_g

=> distance (x_f,f(x_f)) (x_g,g(x_g)) is approx. 0.42759