Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 15



Higher derivatives

The derivative of a differentiable function is also called the first derivative of . The zeroth derivative is the function itself. Higher derivatives are defined recursively.


Definition  

Let denote an interval, and let

be a function. The function is called -times differentiable, if it is -times differentiable, and the -th derivative, that is , is also differentiable. The derivative

is called the -th derivative of .

The second derivative is written as , the third derivative as . If a function is -times differentiable, then we say that the derivatives exist up to order . A function is called infinitely often differentiable, if it is -times differentiable for every .

A differentiable function is continuous due to Corollary 14.6 , but its derivative is not necessarily so. Therefore, the following concept is justified.


Definition  

Let be an interval, and let

be a function. The function is called continuously differentiable, if is differentiable and its derivative is

continuous.

A function is called -times continuously differentiable, if it is -times differentiable, and its -th derivative is continuous.



Extrema of functions

We investigate now, with the help of the methods from differentiability, when a differentiable function

where denotes an interval, has a (local) extremum, and how the growing behavior looks like.


Theorem

Let

be a function which attains in a local extremum, and is differentiable there. Then holds.

Proof  

We may assume that attains a local maximum in . This means that there exists an , such that holds for all . Let be a sequence with , tending to ("from below“). Then , and so , and therefore the difference quotient

Due to Lemma 7.12 , this relation carries over to the limit, which is the derivative. Hence, . For another sequence with , we get

Therefore, also and thus .


We remark that the vanishing of the derivative is only a necessary, but not a sufficient, criterion for the existence of an extremum. The easiest example for this phenomenon is the function , which is strictly increasing and its derivative is zero at the zero point. We will provide a sufficient criterion in Corollary 15.9  below, see also Theorem 17.4 .



The mean value theorem

Theorem

Let , and let

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

Proof  

The statement is true if is constant. So suppose that is not constant. Then there exists some , such that . Let's say that has a larger value. Due to Theorem 11.13 , there exists some , where the function attains its maximum. This point is not on the border. For this , we have , due to Theorem 15.3 .


This theorem is called Theorem of Rolle.

The mean value theorem means that, for every secant, there exists a parallel tangent.

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.


Theorem

Let , and let

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

Proof  

We consider the auxiliary function

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

Hence, fulfills the conditions of Theorem 15.4 , and therefore there exists some , such that . Because of the rules for derivatives, we obtain



Corollary

Let

be a differentiable function such that for all . Then is constant.

Proof  

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.



Theorem

Let be an open interval, and let

be a

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

Proof  

(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.



Corollary

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



Corollary

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.
  1. If holds, then has an isolated local minimum in .
  2. If holds, then has an isolated local maximum in .

Proof


We will encounter a more general statement in Theorem 17.4 .



General mean value theorem and L'Hôpital's rule

The following statement is called also the general mean value theorem.


Theorem

Let , and suppose that

are continuous functions which are differentiable on and such that

for all . Then , and there exists some such that

Proof  

The statement

follows from Theorem 15.4 . We consider the auxiliary function

We have

Therefore, , and Theorem 15.4 yields the existence of some with

Rearranging proves the claim.


From this version, one can recover the mean value theorem, by taking for the identity.

L’Hospital (1661-1704)

For the computation of the limit of a function, the following method called L'Hôpital's rule helps.


Corollary

Let denote an open interval, and let denote a point. Suppose that

are continuous functions, which are differentiable on , fulfilling , and with for . Moreover, suppose that the limit

exists. Then also the limit

exists, and it also equals .

Proof  

Because has no zero in the interval and holds, it follows, because of Theorem 15.4 , that is the only zero of . Let denote a sequence in , converging to .

For every there exists, due to Theorem 15.10 , applied to the interval or , a (in the interior[1] of ,) fulfilling

The sequence converges also to , so that, because of the condition, the right-hand side converges to . Therefore, also the left-hand side converges to , and, because of , this means that converges to .



Example

The polynomials

have both a zero for . It is therefore not immediately clear whether the limit

exists. Applying twice L'Hôpital's rule we get the existence and



Footnotes
  1. The interior of a real interval

    is the interval without the boundaries.


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