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Professor Paul T. Bateman
MONTHLY Problems
Department of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL  61801
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Dear Professor Bateman;

The following is a solution of E3331 showing there is no positive
continuously differentiable function $f$ for which $f'(x) \geq f↑2(x)$
on $[0, ∞)$.

Let $f$ be any positive continuously differentiable function on $[0, \infty)$.
Define $g(x) = x+1/f(x)$, so $g↑\prime(x) = 1 - f↑\prime(x)/f(x)↑2$.  Choose
an arbitrarily large positive number, $a$.  Define $c = g(a) = a+1/f(a) > a$.
Then $g(c) = g(a) + 1/f(c) > g(a)$.  By the mean value theorem, there is some
$b$ in $(a,c)$ for which $g↑\prime(b) = {g(c) - g(a)\over c-a} > 0$, i.e.,
$f↑\prime(b) < f(b)↑2$.  There is no such function $f$ for which $f↑\prime(x) 
\geq f↑2(x)$ on $[0,\infty)$, or even on $[a,\infty)$.

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Sincerely yours,
Robert W. Floyd
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