Sup problem

  • Thread starter andilus
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  • #1
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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.
 

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  • #2
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Look up "Intermediate Value Theorem" or "Bolzano's Theorem."
 
  • #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.
 
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  • #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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