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- #1

- Apr 14, 2013

- 4,036

Could you give me a hint how to prove the following statements?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable (or twice differentiable).

- $\left.\begin{matrix}

\displaystyle{\lim_{x\rightarrow +\infty}f(x)=\ell} \ (\text{or } \displaystyle{\lim_{x\rightarrow -\infty}f(x)=\ell}), \ell\in \mathbb{R} \\

\text{and } f \text{ convex (or concave)}

\end{matrix}\right\}\lim\limits_{\substack{x\rightarrow +\infty \\ (\text{or } x\rightarrow -\infty)}}f'(x)=0 $

$\Rightarrow\ f$ is strictly monotone

- If $\displaystyle{\lim_{x\rightarrow +\infty}f''(x)}$ exists, then $\displaystyle{\lim_{x\rightarrow +\infty}f''(x)=\lim_{x\rightarrow +\infty}f'(x)=0}$ (i.e. we don't assume that $\displaystyle{\lim_{x\rightarrow +\infty}f'(x)}$ exists).