Consider the function
Show that and
, but that the derivative
for all
. Explain why this does not violate Rolle’s theorem.
Proof. First, we show that and
by a direct computation:
Then, we compute the derivative,
To show for any
we consider three cases:
- If
then
implies
(since
times a negative is positive).
- If
then
implies
(since
times a positive is then negative).
- If
, then
is undefined (since
).
Thus, for any
This is not a violation of Rolle’s theorem since the theorem requires that be differentiable for all
on the open interval
. Since
is not defined at
, we have
is not differentiable on the whole interval. Hence, Rolle’s theorem does not apply.