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Sup problem

  1. Oct 15, 2009 #1
    if f is continuous on [a,b] with f(a)<0<f(b), show that there is a largest x in [a,b] with f(x)=0

    i think it can be done by least upper bounds, but i dun know wat is the exact prove.
     
  2. jcsd
  3. Oct 15, 2009 #2
    Look up "Intermediate Value Theorem" or "Bolzano's Theorem."
     
  4. Oct 15, 2009 #3

    LCKurtz

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    As another poster suggested, the intermediate value theorem guarantees there is an x in [a,b] where f(x) = 0. And your idea of using the lub is a good one. So let

    [itex] z = [/itex] lub [itex]\{ x \in [a,b] | f(x) = 0\} [/itex]

    So what you need to show to finish the problem is:

    1. z is in [a,b]
    2. f(z) = 0
    3. No value x > z in [a,b] satisfies f(x) = 0.
     
    Last edited: Oct 15, 2009
  5. Oct 15, 2009 #4

    LCKurtz

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    I now see that this is a duplicate of an identical thread in the Calculus & Beyond section. Please don't do that. It wastes our time answering questions that have already been answered elsewhere.
     
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