[SOLVED] Function without abs. value and showing if its differentiable

**coobe** · September 16th 2009, 05:57 AM

i have to show the following function, without abs. value:

$y=x{{\rm e}^{ \left| x \right| }}$

and then i have to look it its differentiable at the critical point

somehow i must have a mistake here... hard to explain what i did

any help appreciated

**chisigma** · September 16th 2009, 06:20 AM

At $x=0$ is...

$\lim_{h \rightarrow 0+} \frac{(x+h)\cdot e^{x+h}}{h} = \lim_{h \rightarrow 0-} \frac{(x+h)\cdot e^{-x -h}}{h} = 1$

... so that the function is differentiable...

Kind regards

$\chi$ $\sigma$

**VonNemo19** · September 16th 2009, 06:25 AM

Originally Posted by coobe

i have to show the following function, without abs. value:

$y=x{{\rm e}^{ \left| x \right| }}$

and then i have to look it its differentiable at the critical point

somehow i must have a mistake here... hard to explain what i did

any help appreciated

In general, when trying to determine if an absolute value funtion is differentiable, it is useful to determine wether the left and right handed limits are equal

$\lim_{x\to{c^+}}\frac{f(x)-f(c)}{x-c}$

and

$\lim_{x\to{c^-}}\frac{f(x)-f(c)}{x-c}$

**coobe** · September 16th 2009, 07:00 AM

wow thats exactly what i had as a result

i was just thinking i was wrong since i put in the graph in Maple and it looked totally weird at that point...

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