Math 180: Mean Value Theorem (MVT)
Let f be a function such that it is:
1. Continuous on a closed interval [a, b], and
2. Differentiable on the open interval (a, b).
f (b)  f (a )
Then there exists an x-value c, such that f (c) 
. In other words, there exists a point where
ba
the slope of the tangent line at (c, f (c)) is the same as the average rate of change from x  a to x  b
1. Does each function below satisfy the hypotheses of the MVT on the given interval? If so, find all
numbers c that satisfy the conclusion of the MVT. If not, state why it fails the hypotheses.
Interval:
e.
a. [1,5]
b. [–4, 5]
Interval:
c. [–2,3]
d. [2, 4]
f ( x)  x 3  x  1 on the interval [0, 2].
2. A straight highway 270 miles long connects two cities A and B. Prove (using the MVT) that it is
1
impossible to travel from A to B by automobile in exactly 3 hours without having the speedometer
2
register over 75 mi/hr at some point on the trip.