# Math Help - Optimization Problem

1. ## Optimization Problem

I have the following problem:

Maximize with respect to $x$:
$x[1-F(max \lbrace x, \frac{1}{2}x+c \rbrace)],$
where $F$ denotes a CDF and $c$ is an arbitrary constant.

Also, under which conditions should I maximize
$x[1-F(x)],$ or $x[1-F( \frac{1}{2}x+c)],$?

What are the answers?

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

Thank you!

2. ## Piecewise Function and Chain Rule

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

Use the chain rule from here.