Monotone functions – Serlo

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Monotony criterion[Bearbeiten]

The monotony criterion is quite intuitive: if the derivative of a function (i.e. the slope) is positive, it goes up, if the derivative is negative, it goes down. Mathematically, if the derivative of a differentiable function is non-negative or (non-positive) on an interval , then is monotonously increasing (or decreasing) on . If is even strictly positive (or negative) , then is strictly monotonously increasing (or decreasing).

In the first case, even inversion of the statement is true: If a differentiable function is monotonously increasing on , then and if the function is monotonously decreasing on , then then . However, the inversion does not hold true in the strict case, monotone functions do not always have or . For instance, is strictly monotonous, but .

Theorem (Monotony criterion for differentiable functions)

Let be continuous and differentiable on . Then, there is

  1. on monotonously increasing on
  2. on monotonously decreasing on
  3. on strictly monotonously increasing on
  4. on strictly monotonously decreasing on

Proof[Bearbeiten]

The four directions "" follow from the mean value theorem. The two directions "" follow by differentiability of the function:

Proof (monotony criterion for differentiable functions)

We first show the four directions "" and then the two "".

1. : From on we get that in monotonously increasing on .

Let for all and let with . We need to show . By assumption, is continuous on and differentiable on . By the mean-value theorem, there is a with

By assumption, , and hence . Since we have in the enumerator . This is equivalent to , i.e. is monotonously increasing.

2. : From on we get that in monotonously decreasing on .

Let for all and let with . We need to show . By assumption, is continuous on and differentiable on . By the mean-value theorem, there is a with

Now, , and hence . Since we have . This is equivalent to , i.e. is monotonously decreasing.

3. : on implies that is strictly monotonously increasing on

We prove this by contradiction: Let be not strictly monotonously increasing. That means, we have some with and . We need to find a with . Now, is continuous on and differentiable on . So by the mean value theorem, we can find a with

Since , the enumerator of the quotient is non-positive, and because of the denominator is positive. Thus the whole fraction is non-positive, and therefore .

4. : on implies that is strictly monotonously increasing on

Another proof by contradiction: Let be not strictly monotonously decreasing. That means, we have some with and . We need to find a with . Now, is continuous on and differentiable on . So by the mean value theorem, we can find a with

Since , the enumerator of the quotient is non-positive, and because of the denominator is positive. Thus the whole fraction is non-positive, and therefore .

Now, the two directions "" follow:

1. : being monotonously increasing on implies on

Let with . By monotony, . Further, let with . Then we have for the difference quotient

If , then . The enumerator and denominator of the difference quotient are thus non-negative, and so is the total quotient. Similarly in the case of and enumerator and denominator are non-positive. Thus the whole fraction is again non-negative. Now we form the differential quotient by taking the limit . This limit exists because is differentiable on . Furthermore, the inequality remains valid because of the monotony rule for limit values. Thus we have

Since and have been arbitrary, we get on all of .

2. : being monotonously decreasing on implies on

Let again with . By monotony, . Further, let with . Then we have for the difference quotient

If , then and thus the total quotient is non-positive. An analogous statement holds in the case and . By forming the differential quotient we now obtain

Since and have been arbitrary, we get on all of .

Examples: monotony criterion[Bearbeiten]

Quadratic and cubic functions[Bearbeiten]

Example (Monotony of quadratic and cubic functions)

Graphs of the functions and
Graphs of the functions and

For the quadratic power function there is

So is strictly monotonously decreasing by the monotony criterion on and strictly monotonously increasing on .

For the cubic power function there is

So by the monotony criterion, is monotonously increasing on and strictly monotonously increasing on and . The cubic power function is even strictly monotonously increasing on all of .

The fact that with is strictly' monotonously increasing, although only and not , stems from its derivative being zero at only a single point (namely 0). In the end of this article, we will treat a criterion, which tells us when a function is strictly monotonous, even if there is not everywhere .

Question: Why is strictly monotonously increasing on ?

We must show: From with we get . For the cases and we have already shown this with the monotony criterion. So we only have to look at the case . Here there is with the arrangement axioms (missing):

So is strictly monotonously increasing on all of .

Warning

In the example we have seen that the statement " implies strict monotony" does not hold true! This means that from the fact that increases strictly monotonous, we can in general not conclude that . In the example of the function one can also see that the statement " implies strictly monotonous falling" does not hold true in general.

Exponential and logarithm function[Bearbeiten]

Example (Monotony of the exponential and logarithm function)

For the exponential function there is for all :

Therefore, according to the monotony criterion, is strictly monotonously increasing on all of . For the (natural) logarithm function there is for all :

So is strictly monotonously increasing on (not including the 0).

Question: What is the monotonicity behaviour of the logarithm function extended to , i.e.?

There is

Above we have shown that for . So is strictly monotonously increasing on , as well . For on the other hand there is . So is strictly monotonously decreasing on .

Trigonometric functions[Bearbeiten]

Example (Monotony of the sine function)

For the sine function there is

So for all , the is strictly monotonously increasing on the intervals and strictly monotonously decreasing on the intervals .

Question: Where does the cosine function show monotonous behaviour?

Here, .

So for all , the is strictly monotonously increasing on the intervals and strictly monotonously decreasing on the intervals .

Example (Monotony of the tangent function)

For the tangent function there is for all :

Hence, for all , the is strictly monotonously increasing on the intervals .

Question: Where does the cotangent function show monotonous behaviour?

For all , there is

So for all , the is strictly monotonously decreasing on the intervals .

Exercise[Bearbeiten]

Monotony intervals and existence of a zero[Bearbeiten]

Exercise (Monotony intervals and existence of a zero)

Where is the following polynomial function monotonous?

Prove that has exactly one zero.

Solution (Monotony intervals and existence of a zero)

Graph of the function

Monotony intervals:

The function is differentiable on all of , with

So

According to the monotony criterion, is strictly monotonously increasing on and on . Further,

According to the monotony criterion, is strictly monotonously decreasing on .

has exactly one zero:

For , we have the following table of values:

Based on the monotonicity properties and the continuity of that we have previously investigated, we can read off that:

  • On is strictly monotonously increasing. Because of there is for all .
  • On is then strictly monotonously decreasing. So there is also for all .
  • Subsequently increases on again strictly monotonously. Because of and , there must be an with by the mean value theorem. Because of the strict monotony of on , there cannot be any further zeros.

Necessary and sufficient criterion for strict monotony[Bearbeiten]

Exercise (Necessary and sufficient criterion for strict monotony)

Prove that: a continuous function which is differentiable to is strictly monotonously increasing exactly when there is

  1. for all
  2. The zero set of contains no open interval.

As an application: Show that the function is strictly monotonously increasing on all of .

Proof (Necessary and sufficient criterion for strict monotony)

From the monotony criterion we already know that is monotonously increasing exactly when . So we only have to show that is strictly monotonously increasing exactly when the second condition is additionally fulfilled.

: strictly monotonously increasing the set of zeros of does not contain an open interval.

We perform a proof by contradiction. In other words, we show: If the set of zeros of contains an open interval, is not strictly monotonously increasing. Assume there is with for all . Then, by the mean value theorem there is a with

So . If now , then since is monotonously increasing, there is

So there is for all . Hence, is not strictly monotonously increasing.

: the set of zeros of does not contain an open interval strictly monotonously increasing

We perform a proof by contradiction. In other words, we show: if is monotonously, but not strictly monotonously increasing, then the zero set of contains an open interval. Assume there is with with . Because of the monotony of there is

So for all . That means is constant on . Hence there is for all :

so the set of zeros of does contain an open interval and we get a contradiction.

Exercise: is strictly monotonously increasing

is differentiable for all where

as for all . Hence is monotonously increasing. Further there is

So the set of zeros of contains only isolated points, and thus no open interval. Therefore is strictly monotonously increasing on .