# Decreasing $f:\mathbb{R}\to\mathbb{R}$ tending to $0$ at $∞$ not convex beyond any point?

Given a function $$f:\mathbb{R}\to\mathbb{R}$$ differentiable and strictly decreasing such that $$\displaystyle \lim_{x\to\infty}f(x)=0$$, I am looking to find out whether or not there exists an $$x_0$$ such that $$f$$ is convex on $$(x_0,\infty)$$. My guess is that the statement is not true but I can't find a counterexample.

Consider something like $$\frac1x + \frac{\sin(x)}{x^2}+ \frac{4 \sin^2(x/2)}{x^3}.$$ Notice how it keeps wiggling all the way down. Here is its graph from $$100$$ to $$120$$ to give an impression.