Consider the following statement of the intermediate value theorem for derivatives:
Assume is differentiable on an open interval
. Let
be two points in
. Then, the derivative
takes every value between
and
somewhere in
.
- Define a function
Prove that
takes every value between
and
in the interval
. Then, use the mean-value theorem for derivatives to show
takes all values between
and
somewhere in the interval
.
- Define a function
Show that the derivative
takes on all values between
and
in the interval
. Conclude that the statement of the intermediate-value theorem is true.
- Proof. First, since
is differentiable everywhere on the interval
, we know
is continuous on
and differentiable on
. Thus, if
and
then
is continuous at
since it is the quotient of continuous functions and the denominator is nonzero. If
then
hence,
is continuous at
as well. Therefore,
is continuous on the closed interval
. So, by the intermediate value theorem for continuous functions we know
takes on every value between
and
somewhere on the interval
. Since
this means
takes on every value between
and
somewhere on the interval
.
By the mean-value theorem for derivatives, we then know there exists somesuch that
for some
. Since
, we then conclude there is some
such that
for any
. Since
takes on every value between
and
, so does
.
- Proof. This is very similar to part (a). By the same argument we have the function
is continuous on
; thus,
takes on every value between
and
by the intermediate value theorem for continuous functions. Then, by the mean value theorem, we know there exists a
such that
Thus,
takes on every value between
and
. Since
;
takes on every value between
and