Intermediate Value Theorem,

  • Thread starter LilTaru
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  • #1
LilTaru
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Homework Statement



a) Suppose that f(x) is a continuous function on [0, 1] and 0 <= f(x) <= 1 for all x in [0, 1]. Show that there is an x in [0, 1] where f(x) = x.

b) Suppose that f(x) is a continuous function on [0, 2] with f(0) = f(2). Show that there is an x in [0, 1] such that f(x) = f(x + 1).


Homework Equations





The Attempt at a Solution



I assume I have to use the Intermediate Value Theorem, but I have no idea how to use it! For (a) I thought 0 <= f(x) <= 1 means since f(x) is between f(a) and f(b) then there exists a c or x in this question so that f(x) = x, but I have no idea! And for (b)... not a clue! Please help?!
 

Answers and Replies

  • #2
ystael
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Try to construct a new function g from f, in such a way that answering the question means comparing g to a constant (so that it's easier to apply the intermediate value theorem).
 
  • #3
LilTaru
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Oh! Does that mean for (a) g(x) = f(x) - x? Or am I completely off track?!
 
  • #4
ystael
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It's a good thought -- run with it and see where you get to.
 
  • #5
LilTaru
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I am still very confused with how to form g(x)... it is still not making sense how to prove this question!
 
  • #6
Dick
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I am still very confused with how to form g(x)... it is still not making sense how to prove this question!

You already formed a good g(x) by setting g(x)=f(x)-x. If g(0)=0 then f(0)=0 and you are done. Otherwise g(0) is positive, right? What happens if g(1)=0? Suppose g(1) is not zero. What sign is it?
 

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