Chapter 3 Section 2
Rolle’s and Mean Value Theorem
Rolle’s Theorem
Let f be differentiable on (a,b) and continuous on
[a,b]. If f (a )  f (b), then there is at least one
point c belonging to (a,b) where f (c)  0
f (c)  0
a
c
b
(1) Determine whether Rolle’s Theorem can be applied on
the interval, then find the values of c in the interval
2
f
(
x
)

x
 3 x  2 with an interval of
given the function
[1,2]
(2) Determine whether Rolle’s Theorem can be applied on
the interval, then find the values of c in the interval
1
given the function f ( x)  2 x 3  3 x with an interval of [-2,2]
(3) Determine whether Rolle’s Theorem can be applied on
the interval, then find the values of c in the interval
2
given the function f ( x)  3x  2 x with an interval of [2,4]
Mean Value Theorem
• Let f be differentiable
on (a,b) and
continuous on [a,b],
then there exists a
point c belonging to
(a,b) where
f (c )
f (b)  f ( a )
ba
f (b)  f (a )
f (c) 
ba
a
c
b
(4) Find the number which satisfies the MVT for
the function f ( x)  x 2  5 x  7on [-1,3]
(5) Find the number which satisfies the MVT for
the function f ( x)  6  3 on [1,2]
x
(6) Find the number which satisfies the MVT for
the function f ( x)  sin( 2 x)  cos( x) on [0,p]
(7) Suppose the police time you going from one
mile marker to the next in 51.4 seconds, in a 55
mph zone. Do you deserve a ticket?