is continuous on 
andCalculus
Lesson
13 Rolle's Theorem and Mean Value Theorem
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Mean Value Theorem (MVT): if is continuous on and
differntiable on 
,
then there must exist some 
value,
, such that
Rolle's Theorem: The same as MVT, but specialized: when
,
then there must be some 
such that 
(and therefore there
is an extrema there).
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Rolle's Theorem specifically helps with determining whether extrema exist.
MVT is the proof that on some function, there is some point on that function that has
the same slope as the average slope on that interval, and also gives us a method
for finding it.
Example: 
: Find all
values of 
at which the
rate of change
is the same as the average rate of change.
Example: if a police officer tracks your speed over 2 miles at 75mph, there must be
AT LEAST one instant at which your speed was 75 miles per hour.