.

This function is defined on .

The function is the composition of on a rational function; hence it is differentiable on its domain. We have:

The denominator never vanishes and is positive at every point of the domain, thus increases over and over .

The function is a rational function; hence it is differentiable on its domain. We have:

The second derivative vanishes at ; its is negative on the left and positive on the right of this point. Thus the function is concave on and on and is convex on .

Let us now compute the limits of at the open endpoints of its domain.

Noah Dana-Picard 2007-12-28