be
continuous on the closed interval 
and
differentiable on the open interval
.
If 
,
then there is at least one number 
in
such
that 
.Wetzel
Math 140
3.2 – Rolle’s Theorem and the Mean Value Theorem
Rolle’s Theorem –
Let
be
continuous on the closed interval and
differentiable on the open interval.
If ,
then there is at least one number in
such
that .
Ex.:
1.) Find the two intercepts of
and
show that 
at
some point between the two x-intercepts.
The Mean Value Theorem –
If
is
continuous on the closed interval and
differentiable on the open interval ,
then there exists a number c in such
that
Ex.:
2.) 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 (55 mph) at some time during the 4 minutes.