Derivative of |x|

[suresh]

Can someone help me solve the following question

Show that f(x)=|x| is not differentiable at x=0

Thanks

 

[stapel]

Have you looked at the graph of the function? And considered the slopes? And the relationship between slope and derivative?

If not, kindly do so. If yes, then please reply with your thoughts, clarifying where you are stuck.

Thank you.

Eliz.

 

[suresh]

I know it is not differentiable at x=0 because it is a corner and you cannot draw a tangent at this point. Is there a way to mathematically prove it using limits or any other ways

 

[ChaoticLlama]

First of all, I think the proper term is 'cusp' and not 'corner'.

First split your function into piecewise form for the two cases. And then you should find something odd.

 

[Matt]

Can someone help me solve the following question

Show that f(x)=|x| is not differentiable at x=0

Is there a way to mathematically prove it using limits or any other ways

From the definition of the derivative:

\(\displaystyle \L\qquad\begin{eqnarray*}
f'(0)
&=&\lim_{x\to0}\frac{f(x)-f(0)}{x-0}\\
&=&\lim_{x\to0}\frac{|x|-0}{x-0}\\
&=&\lim_{x\to0}\frac{|x|}{x}
\end{eqnarray*}\)

If you can show that the left and right hand limits are different, then the limit does not exist, and hence f'(0) is undefined.