Challanging Problem It is nessecary to solve quicly

[vip89]

Challanging Problem " It is nessecary to solve quicly"

I have a problem :
A car moves along the real line from x = 0 at t = 0 to x = 1 at t = 1, with
differentiable position function x(t) and differentiable velocity function v(t) = x’(t).
The car
begins and ends the trip at a standstill; that is v = 0 at both the beginning and the end of
the trip. Let L be the maximum velocity attained during the trip. Prove that at some time
between the beginning and end of the trip, l v’ l > L^2/(L-1).

Can you verify that L > 1 ?

Thankx

 

[nicksauce]

You can show L > 1 using the Mean Value Theorem

 

[vip89]

By mean value theorem I can find that L>0

 

[nicksauce]

The MVT would say that there exists c in t = [0,1] such that
f'(c) = (x(1) - x(0)) / (t1 - t0) = (1-0) / (1-0) = 1.

 

[vip89]

That is my solution
How I can compleat it??

 

[Hadhod]

You are from UOS
right?