Understanding Absolute Value Sign in Derivative

[sony]

Hi

So I am a little confused about the absolute value sign here. Is the derivative right:

[x-1/|x-1|]e^x + |x-1|e^x

But what do I do when I set this to zero? Do I get two expressions, one for x>1 and one for x<1 ? (thus removing the absolute value signs)

 

[sony]

Hints anyone...?

 

[saltydog]

$$|x-1|=x-1\quad\text{for}\quad x\geq 1$$

and:

$$|x-1|=-(x-1)\quad\text{for}\quad x<1$$

Thus your expression represents this function:

$$y(x)=<br /> \left\{<br /> \begin{array}{rcl}<br /> (x-1)e^x & \mbox{for} & x\geq 1 \\ <br /> -(x-1)e^x & \mbox{for} & x<1<br /> \end{array}\right.$$

and so the derivative will represent two functions likewise but will not exist at x=1. Try plotting this conditional function and you'll see what I mean.