What result tells us that if a biker travels at 45 km in 3 hours,

Question

What result tells us that if a biker travels at 45 km in 3 hours, then her speedometer must have read 15 km/h at least once during the trip? Can we guarantee it will read 12 km/h during the trip? How about 18 km/h?

Answer 1

Let f(x) = the distance the biker has traveled at time x after her start.

f(0) = 0 and f(3) = 45.

Her average speed is $\frac{f(3)-f(0)}{3-0}$ = 45/3 =15

But with the assumption that f is continuous and differentiable, f''(c) = 15 for some value 0<c<3..

Nothing can be said about other speed values unless we know something else about f, e.g. if we know that the max speed was 18 mph, then f will take on every value between 0 and 18.