Rolle's Theorem

Theorem
Let $$f$$ be a real function which is continuous on the closed interval $$\left[{a \,. \, . \, b}\right]$$ and differentiable on the open interval $$\left({a \, . \, . \, b}\right)$$.

Let $$f \left({a}\right) = f \left({b}\right)$$.

Then $$\exists \xi \in \left({a \, . \, . \, b}\right): f^{\prime} \left({\xi}\right) = 0$$.

Proof
Since $$f$$ is continuous on $$\left[{a \,. \, . \, b}\right]$$, it follows from the Continuity Property that $$f$$ attains a maximum $$M$$ at some $$\xi_1 \in \left[{a \,. \, . \, b}\right]$$ and a minimum $$m$$ at some $$\xi_2 \in \left[{a \,. \, . \, b}\right]$$.


 * Suppose $$\xi_1$$ and $$\xi_2$$ are both end points of $$\left[{a \, . \, . \, b}\right]$$.

Because $$f \left({a}\right) = f \left({b}\right)$$ it follows that $$m = M$$ and so $$f$$ is constant on $$\left[{a \,. \, . \, b}\right]$$.

But then $$f^{\prime} \left({\xi}\right) = 0$$ for all $$\xi \in \left({a \, . \, . \, b}\right)$$.


 * Suppose $$\xi_1$$ is not an end point of $$\left[{a \, . \, . \, b}\right]$$.

Then $$\xi_1 \in \left({a \, . \, . \, b}\right)$$ and $$f$$ has a local maximum at $$\xi_1$$.

Hence the result follows from Derivative at Maximum or Minimum‎.


 * Similarly, suppose $$\xi_2$$ is not an end point of $$\left[{a \, . \, . \, b}\right]$$.

Then $$\xi_2 \in \left({a \, . \, . \, b}\right)$$ and $$f$$ has a local minimum at $$\xi_2$$.

Hence the result follows from Derivative at Maximum or Minimum‎.