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Analysis: Derivatives, Rolle's Theorem

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data

    If f has a finite third derivative f''' on (a,b) and if f(a)=f'(a)=f(b)=f'(b)=0 prove that f'''(c)=0 for some c in (a,b)

    2. Relevant equations

    Rolle's Theorem: Assume f has a derivative (finite or infinite) at each point of an open interval (a,b) and assume that f is continuous at both endpoints a and b. If f(a)=f(b) there is at least one interior point c at which f'(c)=0


    3. The attempt at a solution

    Because f is differentiable at a and b, f is continuous at a and b. Also, we have f(a)=f(b)=0

    Therefore, use Rolle's theorem, there exists a<x<e such that f'(x)=0.

    If we know that f' is continuous at a and b (I'M STUCK HERE), then we can conclude that there exists a<y<x such that f''(y)=0 and there exists x<z<b such that f''(z)=0. f''(y)=f''(z). Because f has a finite third derivative on (a,b) (If it is [a,b] then it's done....), f'' is continuous on (a,b) and therefore continuous at y and z. Therefore, by Rolle's theorem, there exists y<w<z such that f'''(w)=0..

    So you know why I am stuck.... Thank your for your help.
     
  2. jcsd
  3. Nov 3, 2012 #2
    My thoughts so far go along the lines of (but I don't really know what I'm talking about):
    f(a)=f(b) so at least one f'(c)=0 on the closed interval [a,b].
    But also f'(a)=f'(b)=0 so we have at least 3 turning points in f'(x) in (a,b).
    Therefore need at least 2 turning points in f''(x) in (a,b)
    Then at least 1 turning point in f'''(x) in (a,b), which is almost the answer.
     
  4. Nov 6, 2012 #3
    By which theorem?
     
  5. Nov 6, 2012 #4

    Dick

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    Mean Value Theorem. superg33k is saying you can show, as you have, that there is a c such that a<c<b such that f'(a)=f'(b)=f'(c)=0. What does that tell you about f'' on (a,c) and (c,b)? What does that tell you about f'''?
     
  6. Nov 7, 2012 #5
    It tells me nothing as f' is not guaranteed to be continuous at a and b so I cannot apply Mean Value Theorem or Rolle's Theorem. Please read my original post.
     
  7. Nov 7, 2012 #6

    Dick

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    If a function is differentiable at a point then it's automatically continuous at that point. You can prove that by looking at the difference quotient.
     
  8. Nov 8, 2012 #7
    But we do not know that f' (NOT f) is differentiable at a and b.
     
  9. Nov 8, 2012 #8

    Dick

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    Ok, I see what you are saying. Finally. The only thing I can think of is to use Darboux's theorem. If f' is discontinuous at 0 then the discontinuity is essential, i.e. the limit x->0 f'(x) doesn't exist. That should make it hard for higher derivatives to be bounded. Seems a little much for a Rolle's theorem problem.
     
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