Rolle's Theorem

If a continuous, differentiable function returns to the same value at both ends of an interval, its derivative is zero somewhere in between.

If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ , then there exists at least one $c \in (a,b)$ with $f'(c)=0$ — a special case of the Mean Value Theorem where the secant slope is already zero.

Examples

For $f(x)=x^2-4x+3$ on $[1,3]$ : $f(1)=f(3)=0$ , so some $c\in(1,3)$ has $f''(c)=0$ — indeed $c=2$ .