most recent 30 from http://mathoverflow.net 2013-05-23T15:26:53Z http://mathoverflow.net/feeds/question/28839 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28839/easy-to-find-roots qwerty1793 2010-06-20T13:14:00Z 2010-06-20T15:27:12Z

<p>Is there a smooth function $f:\mathbb{R} \to \mathbb{R}_{\geq 0}$ such that:</p> <p>1) $\lim_{x \to \infty} = \lim_{x \to -\infty} = 0$</p> <p>2) $\forall x > 0$, $f'(x) &lt; 0$</p> <p>3) $\forall x &lt; 0$, $f'(x) > 0$</p> <p>4) <code>$\forall a_1, \ldots, a_n \in \mathbb{R}, K \in \mathbb{R}_{\geq 0}$</code> the roots of $g(x) = (\sum_{i=1}^n f(x - a_i)) - K$ are "easy to find" (i.e. have an explicit formula in terms of $a_i$ and $K$ for each of them).</p> <p>My initial guesses were $f(x) = \frac{1}{x^2+1}$ and $f(x) = \exp(-x^2)$ but both fail on part 4.</p>