# Homework Help: Mean Value Theorem, Rolle's Theorem

1. Jul 6, 2010

### phillyolly

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

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
f'(t)=g'(t)-h'(t)
g^' (t)-h(t)=0
g'(t)=h'(t)

I am not at all sure I am doing it correctly - my solution is too simple.

2. Jul 6, 2010

### Dick

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.

3. Jul 6, 2010

### phillyolly

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?

4. Jul 6, 2010

### Office_Shredder

Staff Emeritus
He's talking about here, where you seem to assert the derivative is zero for every value of t.

5. Jul 6, 2010

### Dick

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?

6. Jul 6, 2010