Tetration Forum
Question about curvature - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: Question about curvature (/showthread.php?tid=765)

Question about curvature - tommy1729 - 12/15/2012

Let x be a real and consider all f(x) := exp^[1/2](x) that satisfy

1) f(x) is real-analytic.
2) d f(x)/dx > 0 for all x.
3) d^2 f(x) / d^2 x > 0 for all x.
4) f ' (w) = 1 where w is real.
5) let k(x) be the curvature of f at x , then 1/k(x) is U-shaped over the entire real line with its unique minimum value at x=w.

Do all such f(x) agree upon the value of f(-oo) ??

Does adding " d^3 f(x) / d^3 x > 0 for all x > w " create a uniqueness criterion ?

Is there a way to show these conditions to be equivalent to some conditions for sexp(x) ?