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Homework Help: Mean Value Theorem, Rolle's Theorem

  1. Jul 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Two runners start a race at the same time and finish in a tie.
    Prove that at some time during the race they have the same
    speed. [Hint: Consider f(t)=g(t)-h(t), where g and h are
    the position functions of the two runners.]

    2. Relevant equations

    If this is ever helpful:
    Mean Value Theorem:
    f ' (c)= [f(b)-f(a)]/[b-a]

    3. The attempt at a solution

    We know that if a function f is continuous on an interval [a,b] and differentiable on (a,b), and f(a) = f(b) = 0, then there is some point c in (a,b) such that f'(c) = 0.

    The velocity equation is
    g^' (t)-h(t)=0

    I am not at all sure I am doing it correctly - my solution is too simple.
  2. jcsd
  3. Jul 6, 2010 #2


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    You've got it right and too simple but you aren't expressing yourself very well either. The MVT tells you there is a c such that f'(c)=0. It doesn't tell you f'(t)=0 for all t. Explain that to me again.
  4. Jul 6, 2010 #3
    Hi, I don't know how to explain your question. Because I take Calculus I as an online class, I have nobody to explain this to me. May you please help me out to understand this problem?
  5. Jul 6, 2010 #4


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    Gold Member

    He's talking about here, where you seem to assert the derivative is zero for every value of t.
  6. Jul 6, 2010 #5


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    Apply the mean value theorem to the problem. If the start time is a and the finish time is b, what does the mean value theorem tell you? Start with the basics, why are f(a)=0 and f(b)=0?
  7. Jul 6, 2010 #6
    Hi! I got the answer!

    At time t=0, f(start_time) = 0 because at the starting point, both runners are at the same spot. Similarly, at the finishing line, f(finish_time) = 0 because in the end the runners finish tied. Knowing that in the Rolle's theorem at some time c between 0 and the finish time, f'(c) = 0, we can conclude that at some time c, the difference in their velocities is 0, which means that at time c, they essentially have the same speed.
  8. Jul 6, 2010 #7
    Thank you a lot for your support.
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