VIDEO: Computing Monotony - How to Examine Properties of a Function

Basic considerations on monotony behavior

• If you want to calculate the monotony of a function, you must first determine its derivative. To do this, you may need the product, quotient or chain rule, depending on the type of function. You can find these simple rules of derivation in every common formula collection.
• The function is usually divided into individual intervals and a statement is then made as to whether the function is monotonically increasing or decreasing in the observed interval.
• As a result, you must first calculate all extreme points of the function, since the monotony behavior changes at these points.
• Once you have determined all extreme points, consider the intervals between the individual high or low points. Lows.

This is how you can calculate the monotony

After you have calculated the extreme points of the function and divided the function into the intervals described above, you now have to form the derivative f 'of the function. The following then applies to the monotony of the function in the observed interval:

• We have f '(x)> 0, the function is strictly monotonically increasing.
• The following applies: f '(x)> = 0, the function is monotonically increasing.
• We have f '(x) <0, the function is strictly monotonically decreasing.
• The following applies: f '(x) <= 0, the function is monotonically decreasing.

Now calculate the monotony behavior for the other intervals as well.

Calculate monotony - a simple example

Let us consider the function of the normal parabola with f (x) = x².

• The function has only one extreme point, namely the low point T (0 | 0).
• We therefore consider the intervals I.₁=] - ∞, 0] and I₂=]0,∞[
• The derivative of the function is f '(x) = 2x
• So f '(x) <= 0 for x from I.₁ and f thus decreasing monotonically in this interval.
• It is f '(x)> 0 for x from I.₂ and thus f increases strictly monotonically in this interval.
• You can see in each case that the monotony becomes a strict monotony if you omit the interval limits, i.e. the 0 here.

If you use the above instructions for your problems, you can be sure that you will solve your tasks safely and without errors.