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Unique Solution Proof

  1. Apr 17, 2013 #1
    Assume that f is continuous on [a,b] and differentiable on (a,b). Assume also that f′(x) ≠ 0 on (a,b) and f(a) and f(b) have different signs. Show that the equation f (x) = 0 has a unique solution in (a, b).

    I'm not really sure how to even start this proof. Do I need to use the Intermediate Value Theorem? Any help would be great!
  2. jcsd
  3. Apr 17, 2013 #2

    Ray Vickson

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    You actually need to prove two things: (1) there is a solution x in (a,b); and (2) there is only one such solution.

    Do you know how to get (1)?

    For (2): assume the contrary and see what happens.
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