Can you solve this?

Question

Mean value theorem

Suppose two cars start a race at the same time and end it at the same time. Prove that at some point in time, their speeds were the same.

Answer 1

Let f₁(t) be be the distance that the 1st car has travelled by the time t after the start.

f₂(t) be the distance for the 2nd car.

Because the cars start a race at the same time (t=0) and end it at the same time (denote the end time as T)

f₁(0) = f₂(0) = 0, f₁(T) = f₂(T)

Consider a new function g(t) = f₁(t) - f₂(t)

g(0) = 0, g(T) = 0, and g'(t)= f₁'(t) - f₂'(t)

By the mean value theorem there is a moment t*, 0<t*<T, such that

g'(t*) = (g(T) - g(0)) / (T-0) = 0

g'(t*) = f₁'(t*) - f₂'(t*) = 0, so f₁'(t*) = f₂'(t*)

At the moment t* both cars have the same speed.

Answer 2

The mean value theorem states that if a continuous unction f(x) has the coordinates x₁ , f(x₁) and x₂ , f(x₂), there is a point between x₁ and x₂ where the tangent to the function is equal to the slope of the straight line connecting the two points.

If we plotted the distance each car travels as a function of the time, we find that their curves begin and end at the same points. As the instantaneous speed at any time is the tangent to the respective curves, the mean value theorem says that their speeds at some instant of time must be equal to the slope of the straight line connecting the beginning and end points (the times that this happens are not necessarily the same).