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EVT and Fermats to prove f'(c)=0

  1. Dec 7, 2011 #1
    1. The problem statement, all variables and given/known data
    If f is differentiable on the interval [a,b] and f'(a)<0<f'(b), prove that there is a c with a<c<b for which f'(c)=0.


    2. Relevant equations



    3. The attempt at a solution
    Well, I first tried to use IVT but I was having a hard time to I talked to my prof. and he said to use extreme value theorem and fermats theorem. So, by EVT, I know there will be an aboslute maximum and absolute minimum on [a,b]. By f'(a)<0<f'(b) I know that it will be decreasing and increasing and therefore will have a local miniimum on (a,b). Fermats theorem then says that if f(c) is a local extremum, then c must be a critical number of f. Which means the function WILL have a c to make f'(c)=0. I know this intuitively but I'm having a hard time rigorously proving it.
     

    Attached Files:

  2. jcsd
  3. Dec 8, 2011 #2
    no one?
     
  4. Dec 8, 2011 #3

    Mark44

    Staff: Mentor

    Please state the foregoing without "it". What you have written is very unclear. "It" could refer to f'(a), 0, or f'(b). I suspect that none of these is the antecedent.
     
  5. Dec 8, 2011 #4
    Well, I first tried to use IVT but I was having a hard time to I talked to my prof. and he said to use extreme value theorem and fermats theorem. So, by EVT, I know that f will have an aboslute maximum and absolute minimum on [a,b]. By f'(a)<0<f'(b) I know that f will be decreasing and increasing and therefore will have a local miniimum on (a,b). Fermats theorem then says that if f(c) is a local extremum, then c must be a critical number of f. Which means the function WILL have a c to make f'(c)=0. I know this intuitively but I'm having a hard time rigorously proving it.
     
  6. Dec 8, 2011 #5

    Mark44

    Staff: Mentor

    Can you say this more precisely?
    Where will f be decreasing?
    Where will f be increasing?

    If you haven't guessed, I'm trying to get you to make clear, unambiguous statements.
     
  7. Dec 8, 2011 #6
    f will be decreasing on (a,o) and increasing on (0,b)
     
  8. Dec 8, 2011 #7

    gb7nash

    User Avatar
    Homework Helper

    Not necessarily. We only know that the slope of f(x) at a is negative, and the slope of f(x) at b is positive. We don't know the exact intervals f is increasing or decreasing.

     
  9. Dec 8, 2011 #8

    Mark44

    Staff: Mentor

    In addition to what gb7nash said, you are assuming here that a < 0 and b > 0. All you have is the interval [a, b]. There is no indication of whether 0 is in the interval, to the left of it, or to the right of it. It's just some arbitrary interval.
     
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