Rolle's Theorem

It’s a pretty dumb theorem stating the obvious… As per.

Informally, if a continuous differentiable function starts and ends at the same $y$ over a given range, then there’s at least one critical point.

Rolle's Theorem

If:

• Differentiable on a range
• Starts and ends at the same $y$ (I.E. $f(a)=f(b)$)
• $\therefore$ There must be a $c\in (a,b)$ such that $f^{\prime }(c)=0$ (I.E. It’s a extrema…)

Further Applications:

With this, we’re able to generalise more freely to The Mean Value Theorem.