Global maximum of ?(x)-x (consolid'd replies)

[Gene Smith]

On 6/15/06, Joseph Biberstine <jrbibers at indiana.edu> wrote:


> Gene Smith wrote:
> > I think it would be of considerable interest to push this farther.  Has
> > anyone considered using box rather than the ? function? I continue to
> > suspect it would be easier.
>
> Folks keep suggesting Conway's box, but I'm afraid I'm not myself
> schooled enough to see the connection between extrema of f(x)-x and
> x-f_inv(x).


Suppose x = f_inv(y); then f(x) - x = f(f_inv(y)) - f_inv(y) = y - f_inv(y).
Hence the value of the maximum f(x)-x attains at x is the same as what y -
f_inv(y) attains at y. If we find a maximum for y - f_inv(y), we can then
find where f(x) - x attains the corresponding maximum, since x = f_inv(y).
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