# Monotone functions – Serlo

## 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

- on monotonously increasing on
- on monotonously decreasing on
- on strictly monotonously increasing on
- 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)

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)

**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

- for all
- 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 .