Re: Does a simple function exixt with these properties?

In article <dn64ju$m69$1@xxxxxxxxxxxxxxxxxxxxxx>,
israel@xxxxxxxxxxx (Robert Israel) wrote:

> In article <cgulf.32133$tV6.1451@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
> Charles McLane <bitbucket@xxxxxxxxxxx> wrote:
> >Is there a simply-expressible function F(x) defined on [0,infinity) with
> >these properties?
> >
> >F(0) = 0
> >F(infinity) = 1 (in the usual sense of a limit)
> >F(1) = alpha, 0 < alpha < 1
> > (can restrict alpha > 1-exp(-1) if that helps any)
> >dF/dx(0) = 1
> >dF/dx(infinity) = 0
> >dF/dx(x) is monotone and nicely smooth
> >
> >Visualize alpha as "pulling" the curve y(x) = 1-exp(-x) up into the
> >corner between the
> >asymptotes y = 1 and y = x without getting y(x) above either.
>
> Try F(x) = (1 - exp(-x^k))^(1/k) where k = ln(1-exp(-1))/ln(alpha).

Nice idea. I think f(x) = x/(1+x^k)^(1/k) does the same thing,
with k = ln(1/2)/ln(alpha).
.

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