1. ## Optimization Problem

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

Notice $\displaystyle max\{x,\frac12x+c\}$$\displaystyle =\left\{ \begin{array}{rcl} \frac12x+c &\mbox{ if } & x\leq 2c \\x & \mbox{ if } & x>2c \end{array}\right. so \displaystyle x[1-F(max\{x,\frac12x+c\})]$$\displaystyle =\left\{ \begin{array}{rcl} x[1-F(\frac12x+c)] &\mbox{ if } & x\leq 2c \\x[1-F(x)] & \mbox{ if } & x>2c \end{array}\right.$