Suppose that $y=f(x)$ is [[Continuity on a Domain|continuous]] over the closed interval $[a, b]$ and [[Derivative of a function|differentiable]] at every [[Point|point]] of its interior $(a, b)$ If $f(a)=f(b)$, then there is at least one number $c$ in $(a, b)$ at which $f^{\prime}(c)=0$ #### Proof Being continuous, $f$ assumes absolute maximum and minimum values on $[a, b]$ by [[Extreme Value Theorem]]. These can occur only 1. at interior points where $f^{\prime}$ is zero, 2. at interior points where $f^{\prime}$ does not exist, 3. at endpoints of the function's domain, in this case $a$ and $b$. By hypothesis, $f$ has a derivative at every interior point. That rules out possibility (2), leaving us with interior points where $f^{\prime}=0$ and with the two endpoints $a$ and $b$. If either the maximum or the minimum occurs at a point $c$ between $a$ and $b$, then $f^{\prime}(c)=0$ by [[The First Derivative Theorem for Local Extreme Values]], and we have found a point for Rolle's Theorem. If both the absolute maximum and the absolute minimum occur at the endpoints, then because $f(a)=f(b)$ it must be the case that $f$ is a constant function with $f(x)=f(a)=f(b)$ for every $x \in[a, b]$. Therefore $f^{\prime}(x)=0$ and the point $c$ can be taken anywhere in the interior $(a, b)$. --- Useful for showing that a horizontal tangent exists without doing much algebra.