Serlo: EN: Rolle's theorem

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Motivation

Datei:Satz von Rolle – Veranschaulichung und Erklärung.webm We already know from the extreme value theorem that a continuous function $f$ attains a maximum and a minimum on a closed interval $[a,b]$:

The function f is bounded and hence attains a maximum and a minimum

This is of course also true, if $f(a)=f(b)$. In this case (if the function is not constant) there must be a maximum or minimum inside the domain of definition. In the following figure, both the maximum and the minimum are inside $[a,b]$, i.e. within the open interval $(a,b)$:

Special case of the extreme value theorem

Let us now additionally assume that $f$ is differentiable on $(a,b)$. Let $\xi$ be a maximum or minimum. If $\xi$ is inside the domain of definition, i.e. if $\xi is\in (a,b)$, then $f^{\prime }(\xi )=0$ according to the main criterion for extremes values of a differentiable function. This means that the tangent to $f$ at $\xi$ is horizontal. This is exactly what Rolle's theorem says: For every continuous function $f:[a,b]\to ℝ$ with $f(a)=f(b)$, which is differentiable at $(a,b)$, there is an argument $\xi \in (a,b)$ with $f^{\prime }(\xi )=0$.

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  The derivative at the maximum of $f$ is zero.
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  The derivative at the minimum of $f$ is zero.

Of course, $f$ can also assume several (partly local) maxima and minima on $(a,b)$ . Furthermore, it is possible that $f$ attains only one maximum (and no minimum) or one minimum (and no maximum) on $(a,b)$:

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  The function $f$ attains one maximum and no minimum within its domain of definition. At that point, the derivative is zero.
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  The function $f$ attains one minimum and no maximum within its domain of definition. At that point, the derivative is zero.

A special case is $f$ being constant on $[a,b]$. In this case there is $f^{\prime }(x)=0$ for all $x\in (a,b)$:

Sketch of the special case of the extreme value theorem.

This may also happen on a finite sub-interval of $[a,b]$, i.e. on a "horizontal plateau".

No matter which case we looked at, there was always at least one point inside the domain of definition where the derivative of the function is zero.

Rolle's theorem

Datei:Satz von Rolle - Quatematik.webm The theorem named after Michel Rolle (1652-1719) represents a special case of the mean value theorem of differential calculus and reads as follows:

About conditions used in the theorem

There are several necessary requirements in Rolle's theorem. We will show now that if we drop any one of them, the theorem is no longer true.

Application: Zeros of functions

Rolle's theorem can also be used in proofs of existence of zeros. And it can be used to show that a function has at most one zero on an interval. On the other hand, the intermediate value theorem can be used to show that a function has at least one zero on an interval. Together the existence of exactly one zero can be implied.

Outlook: Rolle's theorem and the mean value theorem

As mentioned above, Rolle's theorem is a special case of the mean value theorem. This is one of the most important theorems from real Analysis, as many other useful results can be derived from it. Conversely, we will show that the mean value theorem follows from Rolle's theorem. Both theorems are thus equivalent.

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