Answer
a) Let $f$ be a function that satisfies the following three hypotheses: $f$ is continuous on the closed interval $[a,b]$, $f$ is differentiable on the open interval $(a, b)$, and $f(a)=f(b)$. Then there is a number $c$ in $(a, b)$ such that $f'(c)=0$.
b) Let $f$ be a function that satisfies the following hypotheses: $f$ is continuous on the closed interval $[a,b]$ and $f$ is differentiable on the open interval $(a, b)$. Then there is a number $c$ in $(a, b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$. or, equivalently, $f(b)-f(a)=f'(c)(b-a)$. Geometrically, if one draws a line between the first and last point in the interval, the line has a slope that matches the slope at some point along the line.
