Intermediate Value Theorem: Dr. C. Sean Bohun
Limits and Continuity, Tutorial 07 Page 1
Intermediate Value Theorem
To begin with, let’s start with the basic statement of the theorem.
Theorem: If
f
(
x
) is continuous on a closed interval [
a, b
] and
N
is any number
f
(
a
)
< N < f
(
b
)
then there exists a value
c
∈
(
a, b
) such
f
(
c
) =
N
.
The illustration corresponding to the theo
rem is to the right and indicates that there
may be more than one possible value for
c
.
The important restrictions are that
•
f
(
x
) be continuous and
•
the interval [
a, b
] is closed.
The primary purpose of this theorem is to in
dicate when numbers with various properties
exist.
a
c
b
f
(
b
)
N
f
(
a
)
For example, suppose we have the function
g
(
x
) =
x
2

4
x
and we wish to show there is a number
x
*
such that
g
(
x
*
) = 1.
Step 1)
Notice that since
g
(
x
) is continuous, we can use the intermediate value theorem.
Step 2)
Make a new function
f
(
x
) =
x
2

4
x

1 so that
f
(
x
) = 0 when we have the correct
x
*
.
Note that replacing the function in this manner always makes the
N
in the theorem equal to
zero.
Step 3)
We need to find an
a
and a
b
so that either
f
(
a
)
>
0 and
f
(
b
)
<
0 or
f
(
a
)
<
0 and
f
(
b
)
>
0.
The point here is that we need a change in sign. Choosing