Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inequality of functions

  1. Jan 15, 2017 #1
    Assume f and g are two continuous functions in (a, b).
    If at the start of the segment I've shown f>g by taking the lim where x ---> a+ and the f ' > g ' for every x in (a,b )
    can i say that f >g for all x in (a,b )? is there a theorem for that? that looks intuitively right.
     
  2. jcsd
  3. Jan 15, 2017 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You need to state more clearly what you are assuming.
    You say ##f' > g'## but you only assumed ##f## and ##g## are continuous on ##(a,b)##. Do you mean to assume they are differentiable? And are you saying that ##\lim_{x \to a^+} f(x) > \lim_{x\to a^+}g(x)##? If the answers are yes to these, I suggest you try proving your result using the mean value theorem.
     
  4. Jan 15, 2017 #3
  5. Jan 15, 2017 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Define ##f(a)## and ##g(a)## equal to their right hand limits at ##a## so you can use ##[a,b)## and ##f(a)>g(a)##. Let ##h(x) = f(x) - g(x)##, so you have ##h'(x)>0##. What does the mean value theorem tell you about ##h## if you have a point ##c## in ##(a,b)## where ##f(c)<g(c)##?
     
  6. Jan 15, 2017 #5
    if f(c)<g(c) then h(c) < 0, doesn't help me much
     
  7. Jan 15, 2017 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I asked you what the mean value theorem tells you. Write it down.
     
  8. Jan 15, 2017 #7
    That there is point c where the derivative of h'(c) is parallel to the straight line connecting the two end points of the segment
     
    Last edited: Jan 15, 2017
  9. Jan 15, 2017 #8

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You need to write down the equation the mean value theorem tells you applied to ##h##.
     
  10. Jan 15, 2017 #9
    H'(c)= (h(b)-h(a))/b-a > 0
     
  11. Jan 15, 2017 #10

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Forget the > 0 for a minute. Examine the sign on the left side versus the sign on the right side.
    [Edit, added] I overlooked you haven't used the point c properly on the right side. You don't know anything about h(b). And you don't want c on the left side.
     
    Last edited: Jan 15, 2017
  12. Jan 15, 2017 #11

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Now that the football game I was watching while I was working with you is over, I will be a bit more explicit about what you are doing wrong. Here is what I suggested in post #5:
    Then I asked you to write what the mean value theorem tells you about ##h##. You wrote this:

    There are several things wrong with this besides missing parentheses. First, there is no > 0 in the mean value theorem. Second, you can't use ##c## in ##H'(c)## because I have already used it above where you assume ##c## is a point where ##f(c)<g(c)##. Third, ##f## and ##g## aren't defined at ##b## so neither is ##h(b)##.
    What you need to do is apply the mean value theorem to ##h## on the interval ##[a,c]##. Write it out carefully and completely. You will need to use the words "there exists" somewhere in your application of the mean value theorem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Inequality of functions
  1. An inequality (Replies: 3)

  2. An inequality (Replies: 4)

Loading...