Let $X$ be a nonnegative random variable such that $\mathbb{E}[X] < \infty$.
I am wondering if it possible to characterize the maximizer, $c > 0$ of the following function: $$ f(c) = \mathbb{E}\big[\min\{X, c\}\big]. $$
I know it is equivalent to minimize $-2\min\{X, c\}= 2 \max\{-X, -c\} = - X - c + |X - c|$, hence to minimize $$ g(c) = \mathbb{E}\big[|X - c|\big] - c. $$ The first term is minimized at the median of $X$, however, the second term wants $c \to \infty$.
Based on comments from Mason, we have the monotonicity $$ \min\{X, c\} \leq \min\{X, c'\}, \quad \mbox{a.s.,} $$ whenever $c \leq c'$. Consequently, by a monotone convergence argument: $$ \sup_{c > 0} f(c) = \lim_{c \to \infty} f(c) = \mathbb{E}\Big[\lim_{c \to \infty} \min\{X, c\}\Big] = \mathbb{E}[X]. $$