21this is indeed a nice one......its more a math problem
this is an easy but nice question...
Suppose a car starts from a point A and reaches a point B after time t .The car has definite velocity (which may be variable or constant but it is defined) all through the journey. Let V be the average velocity of the car during this interval. Prove that at some point between A and B the car must have velocity as V (at least once).
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3 Answers
1v_{avg}=\frac{\int_{0}^{T}{v(t)dt}}{T}
then there exist a t in [0, T]
such that
v(t)=\frac{\int_{0}^{T}{v(t)dt}}{T} =v(avg)
its is true for any continous and integrable function
it a fundamental theorem
for more information
http://en.wikipedia.org/wiki/Mean_value_theorem#Proof_of_the_first_mean_value_theorem_for_integration
http://www.sosmath.com/calculus/integ/integ04/integ04.html