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Math Help - Continuous Functions

  1. #1
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    Continuous Functions

    This question seems fairly logical if you think about it as the two functions must intersect. How would you go about proving it?

    Suppose\ that\ f\ and\ g\ are\ continuous functions\ on\ the\ closed\ interval\ [a,b]. Show\ that\ there\ exists\ a\ point\ c \in [a,b] such\ that\ f(c)=g(c)\ if\ f(a) \leq g(a)\ and\ f(b) \geq g(b)
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  2. #2
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    Quote Originally Posted by acevipa View Post
    This question seems fairly logical if you think about it as the two functions must intersect. How would you go about proving it?

    Suppose\ that\ f\ and\ g\ are\ continuous functions\ on\ the\ closed\ interval\ [a,b]. Show\ that\ there\ exists\ a\ point\ c \in [a,b] such\ that\ f(c)=g(c)\ if\ f(a) \leq g(a)\ and\ f(b) \geq g(b)
    Look at h(x)= f(x)- g(x).
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Look at h(x)= f(x)- g(x).
    Sorry, I still don't quite understand.
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  4. #4
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    What is the sign of h(a)? What is the sign of h(b)?
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  5. #5
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    Quote Originally Posted by Defunkt View Post
    What is the sign of h(a)? What is the sign of h(b)?
    Ok I see, so h(c)=0 as intercepts the x-axis. But how would you go about proving it still?
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  6. #6
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    Would you say h(a) \leq 0\ and h(b) \geq 0

    Therefore,\ there\ exists\ a\ c \in (a,b)\ such\ that\ h(a)=0
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  7. #7
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    Yep, that is exactly the intermediate value theorem: Intermediate value theorem - Wikipedia, the free encyclopedia
    Note, though, that since you have a weak inequality, it is possible that c \in [a, b] ie. c=a or c=b.
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