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Rolle's Theorem application

  1. Oct 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Given 2 functions f(x) and g(x) that are differentiable everywhere on R and f′(x) = g(x) and g′(x) = −f(x). Prove that
    1. Between any two consecutive zeros of f(x)=0 there is exactly one zero of g(x)=0,
    2. Between any two consecutive zeros of g(x)=0 there is exactly one zero of f(x)=0.


    2. Relevant equations



    3. The attempt at a solution
    I guess the first question has something to do with Rolle's Theorem but the theorem only states that there exists a zero of f'(x)=0 between 2 zeros of f(x), without mentioning about the uniqueness of that zero. Also I have trouble tackling the second question. Any help is appreciated, thanks!
     
  2. jcsd
  3. Oct 11, 2012 #2

    Dick

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    Suppose there were two zeros of f'(x). What does that tell you about g(x)?
     
  4. Oct 12, 2012 #3
    A contradiction! I think I quite get what you said...

    Let a and b (a < b) be 2 consecutive zeros of f(x)=0, i.e. f(a)=f(b)=0. By Rolle's Theorem, there exists c [itex] \in [/itex] (a,b) such that f'(c)=0, which means g(c)=0.

    Suppose there were 2 zeros of f'(x) between a and b, namely c1 and c2 (a< c1 < c2 < b), then f'(c1)=f'(c2)=0, or equivalently, g(c1)=g(c2)=0. By Rolle's Theorem there exists d [itex] \in [/itex] (a,b) such that g'(d)=0. It follows that -f(d)=0 and thus f(d)=0. This is a contradiction since a and b are 2 consecutive zeros of f(x)=0.
    Therefore there is exactly 1 zero of g(x)=0 between 2 consecutive zeros of f(x)=0.
     
  5. Oct 12, 2012 #4

    Dick

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    Very nice! BTW sin(x) and cos(x) are examples of a pair of functions that have this property.
     
  6. Oct 12, 2012 #5
    Yeah thank you so much!!
    I wonder if there are any non-periodic functions having this property?
     
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