1. Limited time only! Sign up for a free 30min personal 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!

Extreme Value theorem does not hold if [a; b)

  1. Feb 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the statement of the Extreme Value theorem does not hold if [a, b] is replaced
    by [a, b).


    2. Relevant equations



    3. The attempt at a solution

    Please help
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 11, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It's not that hard. Please help us by at least trying.
     
  4. Feb 11, 2012 #3
    i just know that we are going to have open interval and that the will not be either maximum or minimum. But i dont know how to prove it
     
  5. Feb 11, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You don't have to prove anything. You just have to give a counterexample where the extreme value theorem doesn't hold on a half open interval. Please don't PM me. Just reply on the forum thread.
     
  6. Feb 12, 2012 #5
    Given that :

    (Extreme value theorem). If f : [a; b] in R is continuous, then there exist c, d in [a; b] such that

    f(c) = sup{f(x) | x in [a, b]};
    f(d) = inf{f(x) | x in [a, b]}
    Note that since c, d in [a, b], the supremum and infi mum in the above two equations are in fact the maximum and minimum, respectively.

    I tried the follow:

    The function f : [0, 1] given by f(x) = 1 for all x in [0, 1] is continuous and for any
    c,d in [0; 1] we have
    f(c) = 1 = sup{f(x)| x in [0, 1]} = inf{f(x) | x in [0, 1]} = 1 = f(d):

    The function f : [0, 1) in R given by f(x) = x is continuous on [0, 1). If
    S = {x | x in (0, 1)};
    then sup S = 1 and inf S = 0, but these values are not attained. Thus the statement does not hold if [a, b] is replaced by (a, b). 

    Is this counterexample correct?
     
  7. Feb 12, 2012 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, but I think you could say it a lot more simply. f(x)=x on [0,1) has sup(f)=1 but there is no point x in [0,1) such that f(x)=1.
     
  8. Feb 12, 2012 #7
    So just say this?

    The function f : [0, 1) in R given by f(x) = x is continuous on [0, 1).

    then sup(f)=1 ,but there is no point x in [0,1) such that f(x)=1.

    Thus the statement does not hold if [a, b] is replaced by (a, b).
     
  9. Feb 12, 2012 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's good enough for me.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Extreme Value theorem does not hold if [a; b)
Loading...