Analytical Applications of Differentiation · Superchallenge problem

Let $C_1$ be the curve $y=x^2+1$ for $x\ge0$, and let $C_2$ be its reflection in the line $y=x$, namely $y=\sqrt{x-1}$ for $x\ge1$. Find the shortest distance between a point of $C_1$ and a point of $C_2$.