I am wondering if there is possibly a well-known family of functions $f_l:\mathcal{R}\to\mathcal{R}$ parametrized by a single positive real / integer value $l$ that has the following properties:
1). $f_l(x) > f_l(x+\epsilon)$ $\forall l,x,\epsilon > 0$
2). $f_l(l) = 0$ $\forall l > 0$
3). $\int_0^l f_l(x)dx > \int_0^l f_k(x)dx$ $\forall k \neq l $
4). $\lim\limits_{x\to\infty}f_l(x)=-\infty$
Right now I'm using $f_l(x) = \mathcal{N}(x | 0,l) - \mathcal{N}(l|0,l)$, but it doesn't seem to be a principled way, and it does not support the 4th property.
Any links are appreciated!
Thank you!
The straight lines $f_\ell(x) = -x/\ell^2+1/\ell$ seem to meet these criteria.
Indeed, pick any function $g(x)$ that meets the 1st and 4th criteria, and passes through $(1,0)$. Then $f_\ell(x)=g(x/\ell)/\ell$ satisfies the other constraints: All functions will maximize the integral at $\ell$ with value $\int_0^\ell f_\ell(x)=\int_0^1g(x)$, a constant.