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.