Answer
$f(x)=\ln x$ is such a function.
$f$ is concave down on $(0,\infty).$
Work Step by Step
$\ln x$ is such a function.
$\displaystyle \frac{d}{dx}(\ln x)=\frac{1}{x}$ and $\ln 1=0.$
If $f'(x)=\displaystyle \frac{1}{x}=x^{-1}$,
then,
$f''(x)=-x^{-2}$, which is negative for $x\gt 0$.
This means that the graph is concave down on $(0,\infty).$
