1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Differentiability and extreme points question

  1. Sep 6, 2011 #1
    2.b)
    f is continues in [0,1] and differentiable in (0,1)
    f(0)=0 and for [tex]x\in(0,1)[/tex] |f'(x)|<=|f(x)| and 0<a<1
    prove:
    (i)the set {|f(x)| : 0<=x<=a} has maximum
    (ii)for every x\in(0,a] this innequality holds [TEX]\frac{f(x)}{x}\leq max{|f(x)|:0<=x<=a}[/TEX]
    (iii)f(x)=0 for [TEX]x\in[0,a][/TEX]
    (iii)f(x)=0 for [TEX]x\in[0,1][/TEX]
    in each of the following subquestion we can use the previosly proves subquestion.
     
  2. jcsd
  3. Sep 6, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    So what did you try?? If you tell us what you tried, then we'll know where to help?
     
  4. Sep 6, 2011 #3
    i dont want solution only starting guidence.
    for 1 i know the a continues function has a maximum
     
  5. Sep 6, 2011 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    If you want guidance, then you'll have to show us what you've tried. You won't get a solution here...
     
  6. Sep 6, 2011 #5
    there is a difference between solution
    and starting guidence
     
  7. Sep 6, 2011 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    For 1, you indeed use that a continuous function has a maximum. What continuous function do they think they're using here??
     
  8. Sep 6, 2011 #7
    i only know about the function what was given
     
  9. Sep 6, 2011 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    What function do you need to show continuity of??
     
  10. Sep 6, 2011 #9
    f is continues in [0,1] and differentiable in (0,1)
    f(0)=0 and for [tex]x\in(0,1)[/tex] |f'(x)|<=|f(x)| and 0<a<1
     
  11. Sep 6, 2011 #10

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, that is what we are given.

    But we need to prove in (1) that

    [tex]\{|f(x)|~\vert~x\in [0,a]\}[/tex]

    has a maximum. To show this, we must use that every continuous function on a closed interval has a maximum.
    So, what will we take as our continuous function?
     
  12. Sep 6, 2011 #11
    we cant take a spesific function
    its a proof for all function

    a proof for a spesific is not sufficient
     
  13. Sep 6, 2011 #12
    You don't understand what micromass is trying to imply. He's not suggesting taking a specific function like f(x) = sinx. He means you need to choose the correct function in your proof, which has to do with f(x) (hint hint), in order to prove that {|f(x)| | 0<= x <= a} has a maximum.
    The answer is pretty clear - you just need to formulate a short proof.
    Get it?
     
  14. Sep 6, 2011 #13
    ahh now i get it
    we need to formulate some other functionand using it we prove about
    our abstract f(x)

    maybe g(X)=|f(x)|
    ?
     
  15. Sep 6, 2011 #14

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, so prove that |f(x)| is continuous if f is.
     
  16. Sep 6, 2011 #15
    ionly know one way
    and it showing that [tex]lim_{x->x0}f(x)=f(x0)[/tex]

    but its not possible
    because we dont have an actual function
     
  17. Sep 6, 2011 #16

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you show |x| to be continuous??

    Using that, you can show |f(x)| to be continuous as composition of f(x) and |x|.
     
  18. Sep 7, 2011 #17
    You don't need to show that, it's a given.
    Like said above me, asking if the set {|f(x)| | a<=x<=b} has a maximum is just like asking "does the function |f(x)| (notice the absolute value!) get a maximum in the segment [a,b]?". You know there would be an easy answer if you'd say "|f(x)| is continuous on this closed segment and therefore gets a maximum". Therefore, you need to show that |f(x)| is a continuous.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Differentiability and extreme points question
  1. Extreme points (Replies: 0)

Loading...