2 fixed tags
1

# Easy to find roots

Is there a smooth function $f:\mathbb{R} \to \mathbb{R}_{\geq 0}$ such that:

1) $\lim_{x \to \infty} = \lim_{x \to -\infty} = 0$

2) $\forall x > 0$, $f'(x) < 0$

3) $\forall x < 0$, $f'(x) > 0$

4) $\forall a_1, \ldots, a_n \in \mathbb{R}, K \in \mathbb{R}_{\geq 0}$ 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).

My initial guesses were $f(x) = \frac{1}{x^2+1}$ and $f(x) = \exp(-x^2)$ but both fail on part 4.