Marie and Fred are arguing again, this time about the function:

$\displaystyle g \left( x \right) =2\,\arctan \left( {\frac {1+x}{1-x}} \right) +

\arcsin \left( {\frac {1-{x}^{2}}{1+{x}^{2}}} \right) $.

Fred says that if x > 0 then g(x) is a non-constant function whose derivative is zero. Marie says that's nonsense, and even if it weren't, for x < 0, g(x) is a smooth function whose derivative is never zero.

Please help them resolve this dispute.