1. Rolle’s Theorem
and Mean Value
Theorem

2. Theorem:

3. Example
Let f(x) = sin 2x. Find all values of c in the interval
such that f’(c) = 0
Does it satisfy Rolle’s Theorem?
f is continuous
f is differentiable
Find any c values:






3
,
6


f
p
6
æ
è
ç
ö
ø
÷= sin 2×
p
6
æ
è
ç
ö
ø
÷ = sin
p
3
æ
è
ç
ö
ø
÷=
3
2
f
p
3
æ
è
ç
ö
ø
÷= sin 2×
p
3
æ
è
ç
ö
ø
÷ = sin
2p
3
æ
è
ç
ö
ø
÷=
3
2
f
p
6
æ
è
ç
ö
ø
÷ = f
p
3
æ
è
ç
ö
ø
÷
f ' x
( )= 2cos 2x
( )
2cos 2x
( )= 0 cos 2x
( )= 0 2x =
p
2
x =
p
4

4. Another Example:
Let . Find all values of c in the interval
such that f’(c) = 0
Does it satisfy Rolle’s Theorem?
Find any c values:
1,4
( )
f (x)= x2
-5x+4

5. Theorem: Mean Value
Theorem
Let f be continuous on the
closed interval [a, b] and
differentiable on the open
interval (a, b). Then there is
at least one point c in (a, b)
such that
     
a
b
a
f
b
f
c
f




slope of
tangent at c
slope of
secant over
the interval
[a, b]

6. Example
Find a value for c within the interval (-1, 1) where the tangent
line at c will be parallel to the secant line through the
endpoints of the interval.
¢
f x
( )= 3x2
-2x -2
f (1)- f (-1)
1-(-1)
=
-2-0
2
= -1
3x2
-2x -2 = -1
3x2
-2x -1= 0
3x+1
( ) x -1
( )= 0
x = -
1
3
and x =1

7. Another Example
Show that the function satisfies the hypotheses of the
Mean-Value-Theorem over the interval , and find all values of c in the
interval (0, 2) at which the tangent line to the graph of f is parallel to the
secant line joining the endpoints of the interval
f x
( ) =
1
4
x3
+1
0,2
[ ]
f ' x
( ) =
3
4
x2
f 2
( )- f (0)
2-0
=
3-1
2
=1
3
4
x2
=1
x2
=
4
3
x = ±
2
3
x =
2
3
x = -
2
3