I have a log-concave function $f(\cdot)$ that is defined over $\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$. If I know that $f(0)=0$, $\lim_{x \rightarrow \infty}f(x)=0$, and in between it has strictly positive values, then can I conclude that it has a unique maximizer $x^{*}$? and what other properties can I know about this function?
My basic understanding of a log-concave function is that the log of it is concave, so if it starts from zero and ends at zero, then I can only imagine that it will have a unique maximizer, but I can't prove it, and maybe my imagination is flawed.
The function $f(x) = \begin{cases} 0, & x=0 \\ e^{\min(0, 3-({1 \over x}+x)}), & \text{otherwise} \end{cases}$ is log concave and has a maximum value of $1$, but $[1,2] \subset f^{-1} (\{0\})$, so it is not unique.