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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) ?