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

[Gerald McGarvey]

At 04:31 PM 6/16/2006, Gene Smith wrote:


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

Another way to see this is to consider the triangle formed with one side on 
the line x=y, another on the line x = the x value, and another on the line 
y = the y value.
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