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Is it possible to construct a function $f:\mathbb{R} \to \mathbb{R}$ such that there is $c>0$ with the property that for each $n$ and each $x \in \mathbb{R}$ we have $f^{(n)}(x) \geq c$?

If not, is it true than one can for each $n$ find $c_n>0$ such that $f^{(n)}(x)>c_n$ for each $x$?

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The answer to both questions is no. Suppose $f''(x)>c_2>0$. Then $$f'(x_1)-f'(x_0)=\int_{x_0}^{x_1}f''(x)dx>c_2(x_1-x_0),$$ and $$f'(x_0)< f'(x_1)-c_2(x_1-x_0).$$ If you keep $x_1$ fixed and send $x_0\to-\infty$ then $f'(x_0)\to-\infty$.

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  • $\begingroup$ Thank you, simple and proves a lot more, great! $\endgroup$ – truebaran May 11 '16 at 14:58

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