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\leftline{\sevenrm AMM1.Tex[let,rwf]\today}
\leftline{\copyright\sevenrm Robert W. Floyd, September 13, 1989}
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\noindent{\bf AMM Problem E3331, June-July 1989}

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 $[a,\infty)$.
\bye