Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem Let f be continuous on the closed

interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one number c in (a, b) such that f’(c) = 0.

Rolle’s Theorem Rolle’s Theorem does not tell us if c is a

minimum or a maximum Rolle’s Theorem does not tell us the

value of c value where f’(c) = 0.

Rolle’s Theorem only tells us that f’(c) does equal 0 at some point between a & b

Rolle’s Theorem Find the two x-intercepts of f(x) = x2 –

3x + 2 and show that f’(x) = 0 at some point between the two x-intercepts.

Mean Value Theorem

( ) ( )'( )

f b f af c

b a

If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that:

Mean Value Theorem Two stationary patrol cars equipped with

radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit of 55 mph at some time between the two police cars.

Mean Value Theorem

1

60

2

603

60

4

60

miles

hours

1

2

3

4

5

6

5

60