Monotonous Behavior: Investigate & Determine |

In this article we want to explain to you how you can determine the monotony behavior and answer all your questions about it. The monotonic behavior is a topic of the curve discussion and is taught in mathematics.

What is monotonic behavior?

Before we explain to you what monotonic behavior is, you should already have a basic knowledge of differential calculus:

  • What is the first derivative and how do you form it?
  • What is the 2nd derivative and how do you form it?
  • How do I calculate extreme values?

If you know all this, you are well prepared for what follows.

The monotonic behavior of a function tells you in which region the function’s graph rises or falls. Therefore, the monotonic behavior is defined as follows:

  • The function f is strictly increasing if f'(x) > 0.
  • The function f is strictly decreasing if f'(x) < 0.

How do I determine the monotony behavior?

There are two methods you can use to determine the monotonic behavior of a function.

Method 1 for determining the monotonic behavior

The first method requires the second derivative.

  1. Calculate f'(x).
  2. f'(x) = 0, compute zeros of f'(x).
  3. Calculate f»(x).
  4. Insert zeros of f'(x) into f»(x).
  5. name intervals
  6. determine result

Procedure 2 for determining the monotonic behavior

In the second method, you don’t need the second derivative.

  1. Calculate f'(x).
  2. f'(x) = 0, compute NS of f'(x).
  3. name intervals
  4. Draw up a monotony table
  5. Calculate the sign of the intervals
  6. interpret result

You can find out when which method is used in the last section.

Determine monotonic behavior with the second derivative (method 1)

Example 1 Monotonic Behavior (Method 1)

The monotonic behavior of the function f(x) = x² is to be examined.

  1. calculate f'(x)f'(x) = 2x
  2. f'(x) = 0, compute zeros of f'(x)2x = 0 → x= 0
  3. calculate f»(x) f»(x) = 2
  4. Insert zeros of f'(x) into f»(x) f»(x) = 2 > 0 → low point
  5. Naming Intervals The calculated NS divides the relevant range into 2 intervals:
    1. Interval: (-∞; 0)
    2. Interval: (0; +∞)
  6. Determining the resultSince there is a low point at x= 0, the function falls from -∞ to this point.The following applies: (-∞; 0) : strictly monotonically decreasing.To the right of the low point, on the other hand, the function increases.So the following applies: (0; +∞): strictly increasing.

Example 2 Monotonic Behavior (Method 1)

The monotonic behavior of the function f(x) = -x² is to be examined.

  1. Calculate f'(x)f'(x) = -2x
  2. f'(x) = 0, compute roots of f'(x)-2x = 0 → x= 0
  3. calculate f»(x) f»(x) = -2
  4. Insert zeros of f'(x) into f»(x)f»(x) = -2 < 0 → high point
  5. Naming Intervals The calculated NS divides the relevant range into 2 intervals:
    1. Interval: (-∞; 0)
    2. Interval: (0; +∞)
  6. Determining the resultSince there is a high point at x= 0, the function increases from -∞ up to this point.The following applies: (-∞; 0) : strictly increasing monotonically.To the right of the low point, on the other hand, the function increases.So the following applies: (0; +∞): strictly decreasing.

Determine monotonic behavior without the second derivative (method 2)

Example 1 Monotonic Behavior (Method 2)

The monotonic behavior of the function f(x) =x² is to be examined.

  1. calculate f'(x)f'(x) = 2x
  2. f'(x) = 0, compute NS of f'(x) 2x= 0 → x= 0
  3. Naming Intervals The calculated NS divides the range that is relevant to us into two intervals.
    1. Interval: (-∞; 0)
    2. Interval: (0; +∞)
  4. Set up the monotony tableIn the first line of the monotony table you write down the intervals.In the second line you write down the signs of the intervals in step 5.The basic structure of this table then looks like this:

    (-∞; 0)

    (0; +∞)

    f'(x)

  5. Calculate the sign of the intervals To get the sign, you put any number of the interval in the first derivative.
    • From the interval (-∞; 0) we choose the number “-1”:f'(-1) = 2 *(-1) = -2 → negative sign
    • From the interval (0; +∞) we choose the number “1”.f'(1) = 2*1 = 2 → positive sign You write down these intermediate results in the monotony table.
    • (-∞; 0)

      (0; +∞)

      f'(x)

      -+

  6. Interpret resultIf the first derivative of the function in the interval has a positive sign, the graph there is strictly increasing. If the first derivative of the function in the interval has a negative sign, the graph there is strictly decreasing.

    (-∞; 0)

    (0; +∞)

    f'(x)

    sm falling

    +

    sm rising

We have achieved our goal:

We know in which area the graph rises and falls.

Example 2 Monotonic Behavior (Method 2)

The monotonic behavior of the function f(x) = -x² is to be examined.

  1. Calculate f'(x)f'(x) = -2x
  2. f'(x) = 0, compute NS of f'(x)-2x= 0 → x= 0
  3. Naming Intervals The calculated NS divides the range that is relevant to us into two intervals.
    1. Interval: (-∞; 0)
    2. Interval: (0; +∞)
  4. Set up the monotonicity table The intervals are in the first line of the monotonicity table. In the 5th step, you write down the signs of the intervals in the second line. The basic structure of this table then looks like this:

    (-∞; 0)

    (0; +∞)

    f'(x)

  5. Calculate the sign of the intervals To get the sign, you put any number of the interval in the first derivative.
    • From the interval (-∞; 0) we choose the number “-1”:f'(-1) = -2 *(-1) = 2 → positive sign
    • From the interval (0; +∞) we choose the number “1”.f'(1) = -2*1 = -2 → negative sign Write down these intermediate results in the monotony table.

      (-∞; 0)

      (0; +∞)

      f'(x)

      +-

  6. Interpret resultIf the first derivative of the function in the interval has a positive sign, the graph there is strictly increasing. If the first derivative of the function in the interval has a negative sign, the graph there is strictly decreasing.

    (-∞; 0)

    (0; +∞)

    f'(x)

    +

    sm rising

    sm falling

We have achieved our goal:

We know in which area the graph rises and falls.

Special treatment of poles

In the previous examples we only looked at the zeros of the first derivative to determine the intervals. In this context, the poles of a function are used in addition to the zeros.

We’ll show you the whole thing with an example.

Example of special treatment

The function is given.

At x= -1 the function has a pole.

The zeros of the first derivative are: = -2 and = 0.

With this information we can determine the intervals:

(-∞; -2)

(-2; -1)

(-1; 0)

(0; +∞)

f'(x)

To determine the intervals, the poles of a function must be taken into account in addition to the zeros of the first derivative.

Which procedure for determining the monotony behavior is better?

You have now learned two methods with which you can determine the monotonic behavior of a function. But which one is used when?

Pro Method 1 (with second derivative)

If the task also asks about the curvature behavior or turning points in addition to the monotony behavior, use method 1. Because you have to calculate the second derivative one way or another, which is why you can then use it to determine the monotony behavior.

Pro method 2 (without second derivative)

If you don’t need the second derivative in the course of a problem, you can use Method 2 and save time. Because with fractional-rational functions, it can be very time-consuming to determine the second derivative.

Conclusion

The best thing to do is read the entire exercise carefully first and then see whether you need the second derivative at all. This way you can save time and solve the tasks more efficiently.

Monotony behavior – everything important at a glance

We have summarized the most important things for you below:

  • The monotonic behavior shows you how the graph of a function moves in an interval.
  • A graph can be strictly monotonically increasing or strictly monotonically decreasing in an interval.
  • There are two methods you can use to determine monotonic behavior.→ Method 1: when 2nd derivative is needed later.→ Method 2: when 2nd derivative is not needed later.

Well done! After diligently reading through everything, you should now know everything about monotony behavior. 🙂 Keep it up!