Two runners start a race at the same time and finish in a tie. Prove t

vakirnarhh 2021-11-16 Answered
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 and are the position functions of the two runners.
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Expert Answer

Steacensen69
Answered 2021-11-17 Author has 15 answers
Step 1
f(t)=g(t)h(t) - the difference positions of two runners.
g(t), h(t) are continuons and differentiable. So, f(t) is too continuons and differentiable.
We are attempting to prove that at some point during the race the difference in velocities between the runners was 0.
Find the first derivative of the equation f(t)=g(t)h(t).
f(t)=g(t)h(t)
Consider the intreval (t0,t1), where is the start time t0 and the finish time t1.
Is given that runners start a race at same time and finis in a tie.
So, g(t0)=h(t0) and g(t1)=h(t1).
We have: f(t0)=0 and f(t1)=0
We can apply the Rolle's theorem, because we have: f(t0)=f(t1) and f(t) is continuons and differentiable.
So, there exists some cc that such f(c)=0
f(c)=g(c)h(c)=0
g(c)=h(c)
For c=t, t(t0,t1)
g(t)=h(t)
Answer
Apply the Rolle's theorem.
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