Dear readers,

I have the following problem:

Maximize with respect to$\displaystyle x$:

$\displaystyle x[1-F(max \lbrace x, \frac{1}{2}x+c \rbrace)],$

where $\displaystyle F$ denotes a CDF and $\displaystyle c$ is an arbitrary constant.

Also, under which conditions should I maximize

$\displaystyle x[1-F(x)],$ or $\displaystyle x[1-F( \frac{1}{2}x+c)],$?

What are the answers?

This has been bugging me for a while, please help.

Thank you!