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Math Help - question in sup of two functions...

  1. #1
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    question in sup of two functions...

    f,g: [a,b]---> R
    f(x)>g(x) for every x at [a,b]

    which of these statements are true? and why?
    1. if f and g are continous at (a,b), and f is bounded at [a,b] - so sup f((a,b))>sup g((a,b)).
    2. if f and g are continous at [a,b], so sup f([a,b])>sup g([a,b]).

    thanks
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  2. #2
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    Re: question in sup of two functions...

    Quote Originally Posted by orir View Post
    f,g: [a,b]---> R
    f(x)>g(x) for every x at [a,b]
    which of these statements are true? and why?
    1. if f and g are continous at (a,b), and f is bounded at [a,b] - so sup f((a,b))>sup g((a,b)).
    2. if f and g are continous at [a,b], so sup f([a,b])>sup g([a,b]).

    For #1, let [a,b]=[0,1] and consider f(x)=x^3~\&~g(x) = \left\{ {\begin{array}{*{20}{rl}}  {{x^2},}&{0 < x < 1} \\   { - 1,}&{x \in \left\{ {0,1} \right\}} \end{array}} \right.
    Can you show that \sup(f([0,1])=\sup(g([0,1])~?


    For #2, think High point theorem.
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  3. #3
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    Re: question in sup of two functions...

    thanks..
    but, what is High point theorem?
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  4. #4
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    Re: question in sup of two functions...

    Quote Originally Posted by orir View Post
    what is High point theorem?

    If f is a continuous function on [a,b], then \exists h\in [a,b] such that f(h)=\sup(f(a,b])).
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  5. #5
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    Re: question in sup of two functions...

    ok! so using this theorem 2# is easy.. thank you!
    how can i find more details on this theorem? i couldn't find it by just searching "Hight point theorem" up... are there anymore usefull sentences like this one?
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