Question

Consider a car traveling on a highway. If the car travels 100 miles in 2 hours, which theorem guarantees that the car must have been traveling at 50 mph at some point in those two hours? (You may assume position and velocity are continuous and differentiable)

Answer 1

The answer to this question is MEAN VALUE THEOREM

As it states,

Let be a continuous function on the closed interval , and differentiable on the open interval , where . Then there exists some in such that

Therefore, as the displacement is a function of time displacement X can be written as X(t).

Now, as per mean value theorem velocity v can be written as

Now putting the given values   

Where x(t₂) is displacement at time t₂=2 hours, which is 100 miles, x(t₁) is displacement at time t₁=0 hours,which is 0.

So the mean value theorem guarantees that the car must have been traveling at 50 mph at some point in those two hours.

Please comment for further help.