Rolle's Theorem

Rolle's Theorem Definition

Now that we've gone over the conditions for Rolle's Theorem, let's look at what this theorem says.

Rolle's Theorem states that if a function $f$ is:

• continuous on the closed interval $[a,b]$
• differentiable on the open interval $(a,b)$
• $f\left(a\right)=f\left(b\right)$

then there exists at least one number $c$ in $(a,b)$ such that $f\text{'}\left(c\right)=0$.

Geometrically speaking, if a function meets the requirements listed above, then there is a point $c$ on the function where the slope of the tangent line is 0 (the tangent line is horizontal).

A continuous and differentiable function f that has points a and b such that f(a) = f(b) has at least one point c where the slope of the tangent line is 0 - StudySmarter Original

In our walking example, Rolle's Theorem says that since we started and ended at the same place, there must have been a movement where we made a turn (the derivative is 0).

Rolle's Theorem vs. The Mean Value Theorem

Recall the Mean Value Theorem, which states that if a function $f$ is:

• continuous on the open interval $(a,b)$
• differentiable on the closed interval $[a,b]$

then there is a number c such that $a<c<b$ and

$f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Rolle's Theorem is a "special case" of the Mean Value Theorem. Rolle's Theorem says that if the requirements are met and there are points a and b such that $f\left(b\right)=f\left(a\right)$, or $f\left(b\right)-f\left(a\right)=0$, then there is a point $c$ where $f\text{'}\left(c\right)=0$. If we plug in $f\left(b\right)-f\left(a\right)=0$ to the Mean Value Theorem equation for $f\text{'}\left(c\right)$, we get $f\text{'}\left(c\right)$. So, Rolle's Theorem is the case of the Mean Value Theorem where $f\left(b\right)=f\left(a\right)$.

Rolle's Theorem Proof

Rolle's Theorem Step-by-Step Procedure

While no explicit formula is associated with Rolle's Theorem, there is a step-by-step process to find the point $c$.

1. ensure that the function meets Rolle's Theorem: continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$.

2. plug a and b into the function to guarantee that $f\left(a\right)=f\left(b\right)$.

3. If the function meets all requirements of Rolle's Theorem, then we know that we are guaranteed at least one point $c$ where $f\text{'}\left(c\right)=0$.

4. To find $c$, we can set the first derivative equal to 0 and solve for $x$.

Rolle's Theorem Examples

Example 1

Example 2