Rolle's theorem

Theorem

Suppose real-valued function $f$ is continuous on proper closed interval $[a,b]$, differentiable on open interval $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = 0$ (i.e. $f$ has a critical point in $(a,b)$).

Notes

• may be considered a special case of mean value theorem
• see Taylor's theorem

References

1. https://en.wikipedia.org/wiki/Rolle's_theorem
2. https://tutorial.math.lamar.edu/classes/calci/MeanValueTheorem.aspx