1. ## Graph Transformation Problems

1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.

Thanks a lot

-aussiekid90

2. Originally Posted by aussiekid90
1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.
Consider the general absolute value $y=|x|$.
Is it differenciable at $x=0$?
No.

Because, at $x=0$ we have.
$\lim_{\Delta x\to 0}\frac{|\Delta x|}{\Delta x}$
If we approach from the right we have 1.
If we approach from the left we have -1.
The sided limits do not conincide.

3. G'day, aussiekid90!

1. If $f(x)\:=\:x^2-2x$, is $|f(x)|$ differentiable at $x=0$?
The graph of $f(x) \:=\:x^2-2x$ looks like this:
Code:
            * |               *
|
*|              *
--------*-------------*--
0| *         * 2
|    * *
|
In the graph of $g(x) \:=\:|x^2-2x|$,
. . anything below the x-axis is reflected upward.
Code:
            * |               *
|    * *
*| *       *  *
--------*-----------*--
|
The graph has two different slopes at $x = 0$.
. . The function is not differentiable there.

2. For the same curve, is $f(|x|)$ differentiable at $x=0$?

The graph of $f(|x|)$ is symmetric to the y-axis.
The graph looks like this:
Code:
      *               |               *
|
*              |              *
--*-------------*-------------*--
*         * | *         *
* *    |    * *
|

The graph has two different slopes at $x=0$.
. . The function is not differentiable there.