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Determine whether the following statement is true or false. If $f'(x)<0$, for all real numbers $x$, then $f$ decreases without bound.

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2 Answers 2

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False. Consider $f(x)=e^{-x}$. Observe that $f'(x)=-e^{-x}<0$ for all $x\in \mathbb{R}$. Yet since: $$ \lim_{x\to\infty} e^{-x}=0 $$ we know that $f$ is decreasing but is bounded below by $0$.

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  • $\begingroup$ Thank you!!! you are a STAR. $\endgroup$
    – Sara Sharp
    Jun 26, 2013 at 0:59
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Hint: Consider this plot of $f(x)=e^{-x}$:

enter image description here

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