here my question:
Suppose $Y$ is a random variable distributed on $[0,1]$ accodring to cdf $F(Y).$ Assume cdf $F(Y)$ admits a differentiable density $f(Y).$ Let $a \in [0,1]$ be a fixed parameter. Is there any class of distributions or any further assumptions that guarantee that the following problem is concave:
$$\max F(y) (a-y)$$
where one maximizes over $y \in [0,1]$.