Two Calculus Questions.

[DJBruce]

Here are the two questions I am stuck on:

 

"A real valued function has the property that the absolute value of the function is differentiable at o. Show by example that the function need not be differentiable at 0. Prove that if we require the function to be continuous at 0 then the function must be differentiable at 0."

 

For the first part I was thinking of:

 

[math] f = x^2+2x [/math] for [math]x \leq 0 [/math] and [math] -x^2-2x [/math] for [math] x > 0 [/math]

 

Looking at the derivative of f at 0 we see that the LHD=2, while the RHD=-2, so the function is not differentiable at 0. However, looking at the absolute value of the f we see we have the derivative is 2 for both sides so it is differentiable. However, checking to see if this function is continuous I find that it is, but this would contradict the second statement on of the problem. What is my mistake?

 

"Let g be a real valued function defined as 0 when x=0, and exp([math]-x^{-2}[/math]) when [math] x \neq 0 [/math]. Prove that g is cinfinity at 0, and that the kth derivative at 0 is 0. "

 

I am not sure where to start with this proof, I was considering induction, bit I am not sure about that. So could anyone give me any ideas.

 

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So I just realized by idiotic error in the first question. Sorry for that.

 

I am, however, still unsure on the second question.

Edited November 28, 2010 by DJBruce