Page 1 of 2 12 LastLast
Results 1 to 15 of 25

Math Help - Analysis Problem dealing with derivatives and neighborhoods

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    22

    Analysis Problem dealing with derivatives and neighborhoods

    1. Let f be a function defined on some neighborhood of x=a. Prove f '(a)=0 iff a is in D(abs. value f)

    show me a nice proof..

    thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by antoniowilliams46 View Post
    1. Let f be a function defined on some neighborhood of x=a. Prove f '(a)=0 iff a is in D(abs. value f)

    show me a nice proof..

    thanks in advance
    What is D(|f|)?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    22
    D means differentiable
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by antoniowilliams46 View Post
    D means differentiable
    So you're question is show that f'(a)=0 if and only if |f| is differentiable at a?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    22
    f'(a)=0 iff a, then element sign, D(abs. val. of f)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by antoniowilliams46 View Post
    f'(a)=0 iff a, then element sign, D(abs. val. of f)
    That doesn't answer my question.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Apr 2010
    Posts
    22
    the answer to your question is yes... sry
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    If you mean that {\mathrm{d}f\over\mathrm{d}x}(a) =0 iff {\mathrm{d}|f|\over\mathrm{d}x}(a) = 0, try using the backwards triangle inequality \Big||a|-|b|\Big|\le|a-b|.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Apr 2010
    Posts
    22
    how would i go about doing that
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    \Big|{|f(a+h)|-|f(a)|\over h}\Big| \le \Big|{f(a+h)-f(a)\over h}\Big| so if f' is 0 at a |f|' is zero at a too. The other direction is false, i.e. if |f| is differentiable at a, f could fail to be differentiable at a. I'm still not sure what you mean by D though, so I'm not sure if this is what you want...
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Apr 2010
    Posts
    22
    D(f)={x:f'(x) exists}
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Newbie
    Joined
    Apr 2010
    Posts
    22
    so what do we do now
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    Let f = 1 on irrationals and -1 on rationals. Then f' does not exist anywhere, but |f| is differentiable everywhere. So a\in D(|f|) does not imply f'(a)=0.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Newbie
    Joined
    Apr 2010
    Posts
    22
    so how would i prove the originial problem?
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    Well the original problem appears to be wrong. If you mean " f'(a)=0 iff |f| is differentiable at a" then it is false by the example I gave in post #13. If you mean "if f'(a)=0 then |f| is differentiable at a (with derivative 0)" then I see post #10.
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. derivatives 2/analysis
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 16th 2011, 02:52 PM
  2. A problem dealing with Vectors
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: August 26th 2011, 02:35 PM
  3. Real Analysis Proof help dealing with continuity
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: April 11th 2010, 09:22 PM
  4. analysis of derivatives
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 23rd 2009, 03:33 PM
  5. problem dealing with 1^k + 2^k + ... +n^k
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: January 19th 2009, 06:56 AM

Search Tags


/mathhelpforum @mathhelpforum