# Differentiable functions/Mean value theorem/Section

Let , and let

be a continuous function, which is differentiable on , and such that . Then there exists some , such that

This theorem is called *Theorem of Rolle*.

The following theorem is called *Mean value theorem*. It says that if a function describes a differentiable one-dimensional movement, then the average velocity is obtained at least once as the instantaneous velocity.

Let , and let

be a continuous function which is differentiable on . Then there exists some , such that

We consider the auxiliary function

This function is also continuous and differentiable in . Moreover, we have and

Hence, fulfills the conditions of fact, and therefore there exists some , such that . Because of the rules for derivatives, we obtain

If is not constant, then there exists some such that . Then there exists, due to the mean value theorem, some , , such that , which contradicts the assumption.

Let be an open interval, and let

be a

differentiable function. Then the following statements hold.- The function is increasing (decreasing) on , if and only if () holds for all .
- If holds for all , and has only finitely many zeroes, then is strictly increasing.
- If holds for all , and has only finitely many zeroes, then is strictly decreasing.

(1). It is enough to prove the statements for increasing functions. If is increasing and , then the difference quotient fulfills

for every with
.
This estimate carries over to the limit as , and this limit is .

Suppose now that the derivative is . We assume, in order to obtain a contradiction, that there exist two points
in with
.
Due to the
mean value theorem,
there exists some with
and

which contradicts the condition.

(2). Suppose now that
holds with finitely many exceptions. We assume that
holds for two points
.
Since is increasing, due to the first part, it follows that is constant on the interval . But then
on this interval, which contradicts the condition that has only finitely many zeroes.

A real polynomial function

of degree has at most local extrema, and one can partition the real numbers into at most intervals, on which is alternatingly strictly increasing or strictly decreasing.

### Proof

Let denote a real interval,

a twice continuously differentiable function, and an inner point of the interval. Suppose that

holds. Then the following statements hold.- If holds, then has an isolated local minimum in .
- If holds, then has an isolated local maximum in .

### Proof

We will encounter a more general statement in
fact.