You want to build a house by a river. In summer you take water from the river for your garden pond. Unfortunately, this is only possible as long as the river carries enough water. In addition, of course, you do not want your house to be flooded during high water, so you build a small dam.
Wouldn’t it be cool if you had a function that describes the water level of this river where you live?
The function actually describes the water level of the river within a year in the interval, where x is the time in months and the height of the river in meters. You could now use this function to see when the river has the least water and when it has the most water, and how much water there is in the river at each time.
You can find out exactly how to do this in this explanation.
Determine extreme points
Low and high points are extreme points. They each represent the lowest and highest point in the vicinity.
If there is a low or a high with , then the x value this point too extreme point called.
Of the y value this point will also extreme value called.
But what criteria must be met for a low and a high point?
Necessary condition for an extreme point
In order to check the necessary condition, the derivative of a function is first formed.
For the derivation of a function at the point, the following applies as a necessary criterion for an extreme point:
Check the necessary condition on the input example.
The necessary condition on the function is to be checked with . For this, the derivation is required first.
Now set this derivative equal to 0.
Since it is later advantageous to continue calculating with the exact x-value, the square root is sometimes taken at this point.
The necessary condition is therefore satisfied for and .
The necessary condition is not enough to say that an extreme point exists at the point.
Sufficient condition for an extreme point
In addition to the necessary condition, the sufficient condition must be met.
For the second derivative of a function at the point, the following is a sufficient criterion for…
… a low point:
… a peak:
Another way to determine a low or high point is to look at the first derivative at . If there is a sign change from + to – (corresponds to ), then there is a high point at this point. If there is a sign change from – to + (corresponds to ), then there is a low point at this point.
This procedure is usually useful when a diagram is given or the formation of the second derivative is significantly more complex.
Now check the sufficient condition for the input example.
The graph of the function with looks like this:
Figure 2: Graph of a 4th order function
It can be seen that the function has an extreme point at .
First form the first and second derivative.
Next, apply the necessary criterion.
Now try to apply the sufficient criterion.
In this case, the second derivative is also equal to 0. Here the possibility of looking at the change of sign of the first derivative at the point comes into play. Look at the diagram of the first derivative.
Figure 3: Diagram of a derivative function
In the diagram it can be seen that there is a change of sign from – to + at this point. Alternatively, the sign change can be determined as follows.
This calculation also makes it clear that there is a change of sign from – to + at this point. It is therefore a low point at the point.
e function
The e-function represents another special case with regard to extreme points.
Around extreme points the e function to calculate, you would first have to calculate the zeros of the first derivative.
So the expressions 0 can be must be one of the factors be. the parameter b and c are defined in such a way that they are not 0 may be. Accordingly would have to etc the value 0 correspond to.
Since you already know that the pure and the extended e-function have no zeros, ecx cannot either 0 be.
Thus the e-function has none extreme pointsso neither a high point nor a low point, and no turning points.
saddle point
Another possibility when the second derivative is 0 is a so-called saddle point. However, since this is a turning point and therefore NOT an extreme point, it is only briefly considered here.
To get a better idea of what a saddle point is, you can look at the figure below.
Figure 4: Graph of a saddle point
Now look at the whole thing again mathematically.
If for the job
holds, a saddle point exists. This point is NO extreme pointbut a turning point.
In the case of a saddle point, there is NO sign change at the point of the first derivation, but rather a double zero.
Extreme Points – Calculate high points and low points
Since you already know the necessary and sufficient criterion for extreme points, you can use the following recipe to calculate extreme points.
Local and global extreme points
At extreme points, a distinction is not only made between low and high points, but also between local and global extreme points.
If there is a low point , then there is a local minimum.
If there is a high point , then there is a local maximum.
Great, now you know that at every extreme point, , is a local extremum. But then what is a global extremum?
The largest function value that a function can assume in the domain of definition is also referred to as the global maximum.
The smallest function value that a function can assume in the domain of definition is also called the global minimum.
This can either be the y-value of the highest high point/lowest low point or be related to the global course and thus also be or. You can always use the value range to help you with this.
In this case, you can also look at the introductory example.
Extreme places – exercise
Finally, take care of the interpretation of the input example.