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Suppose a positive and strictly monotone increasing differentiable function $f: \mathbb R \rightarrow \mathbb R $ is $O(g(x))$ for some function $g$ that is strictly positive for large $x$, and suppose $g$ is $O(x)$ (i.e. linear growth or slower). Also suppose $\lim_{x \to \infty}f(x) \rightarrow \infty$ and $\lim_{x \to \infty}f'(x) \rightarrow 0$. I am trying to determine whether $x f'(x))$ is also $O(g(x))$.

Jong
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  • Definitely not. You can add $\sin(x^3)$ (say) to $f(x)$ without altering the hypotheses, but that affects $f'(x)$ enormously. – Greg Martin Aug 14 '22 at 05:57
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    But that would alter the hypotheses that f is strictly monotone – Jong Aug 14 '22 at 15:11
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    Right, I missed that assumption (and now a new one has been added as well). One can still perturb a nice function like $f(x)=g(x)=\sqrt x$ so that it has small little local regions with as large a derivative as you'd like (so for example, at every integer $n$ make it have derivative $n^{-1/4}$ which is larger than "expected"). So the answer is still going to be no. It's very rare for bounds on a function's values to imply bounds on the function's derivative. – Greg Martin Aug 14 '22 at 16:38

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