2. Rolle’s Theorem and Mean
Value Theorem
Rolle’s Theorem

Let f be a function that satisfies 3 hypotheses:





f is continuous on the closed interval [a,b]
f is differentiable on the open interval (a,b)
f(a) = f(b)
Then there is a number x=c in (a,b) such that
f’(c)=0
This makes sense because if a function has the
same y value in 2 places, it must turn around
somewhere in between and therefore the
derivative will be 0
Rolle’s Theorem

Here are some functions that satisfy Rolle’s
Theorem
Application

Rolle’s Theorem is often applied with motion
problems such as throwing a ball into the air.
You know that sometime in between when the
ball is thrown and when it returns to the
ground, it must turn around and therefore its
derivative will equal 0 at its maximum height.
Example 1

Determine if Rolle’s Theorem applies. If so,
find the values of x guaranteed by the theorem
for f ( x )  x  2 x on (  2, 2)
Note: Even if you given an open interval,
when the theorem is invoked, we MUST
check the conditions on a closed interval.
4

2
Mean Value Theorem

French mathematician Joseph-Louis Lagrange
first stated the next important theorem.
Mean Value Theorem
Connects concepts of average rate of change
over an interval (secant line) with the
instantaneous rate of change at a point within
that interval (tangent line)
 Somewhere between
points a and b on a differentiable curve,
there is at least one
tangent line parallel
to secant AB.

Mean Value Theorem (MVT)

If f(x) is continuous at every point on [a,b]
and differentiable on (a,b), then there is at
least one point c in (a,b) where
f '( c ) 
Slope of tangent line

f (b )  f ( a )
ba
Slope of secant line
Rolle’s Theorem is the special case of the
MVT where f(a) = f(b)
Example 2

Show that f(x) = x2 satisfies the Mean Value
Theorem on [0,2]. If so, find the value(s)
guaranteed by the theorem.
Example 3



Why does the mean value theorem not work
on [-1,1] for the following functions?
Not differentiable at x=0
x 1
 x  3,
 2
 x  1,
2
x  1 Not continuous at x=1
x 1
Example 4

Calculator permitted. Determine all the
numbers c which satisfy the conclusion of the
MVT for the function f ( x )  x  2 x  x on [-1,2].
3
2
Example 5
The calculus cops have set up their elaborate speed trap on a busy stretch of road.
A suspected pumpkin farmer who uses the road daily to haul his pumpkins to
market is suspected of chronic speeding well beyond the posted limit of 55 mph.
The calculus cops aim to finally ticket this unlawful transporter of seasonal
gourds. The calculus cops set up 5 miles apart from each other. Calculus cop A
spots the farmer with his truck loaded down with would-be jack-o-lanterns. As
the farmer passes cop A, he is clocked at a paltry 50 mph. Cop A could have
sworn the farmer waves at him as he drives by. Cop A immediately radios
Calculus cop B, whereby cop B starts his timer. Four minutes later, cop B clocks
the farmer cruising by at only 55 mph. He clearly sees the farmer wave at him
with a giant grin that would make a jack-o-lantern jealous. After a quick
calculation on his field-issued TI-84 calculator, he pulls out with his lights on to
issue a speeding ticket to the ornery pumpkin farmer. For what speed can
calculus cop B ticket the farmer? Would this ticket hold up in court? Why or
why not?
Example 6

Let f(x) be a function that is differentiable for
all x.


a. Suppose that f(0)=-3 and f’(x)≤ 5 for all values
of x. How large can f(2) possibly be?
b. Suppose that f(5)=2 and f’(x)≥ -3 for all values
of x. How large can f(1) possibly be?
MVT

The MVT can help establish some basic facts
of differential calculus, such as

If f’(x)=g’(x) for all x in an interval then f and g
are “parallel” on that interval