Question

An automobile travels 4 miles road in 5 minutes. Use the Mean Value theorem to show that the speedometer reads exactly 48 mph at least once during the trip.

Answer 1

MVT states that for an interval [a,b] there is a c such that f'(c) = (f(b)-f(a))/(b-a)

Here letting f(x) be distance at time x minutes,
f(0) = 0
f(5) = 4

(f(5)-f(0))/(5-0) = 4/5 mi/min (that's 48 mph)

MVT states that at some 0<= c <= 5, f'(x) = 4/5

f'(x) is the speedometer reading at x minutes.