You are doing your homework and are supposed to determine all the inflection points of the following function.
You look helplessly on the internet for support. You sigh in relief because you encounter this explanation. Your homework is saved.
Curve Discussion – Calculate inflection points of a function
Inflection points are points at which the curvature behavior of a function changes.
If there is an inflection point with , then the x value this point too turning point called.
Of the y value this point will also turning value called.
But what criteria must be met for a turning point?
Calculate turning point – Necessary condition
In order to check the necessary condition, the second derivative of a function is formed.
For the second derivative of a function at the point, the necessary criterion for an inflection point is:
Check the necessary condition on the input example.
The necessary condition is not enough to say that a turning point exists at the point.
Calculate turning point – Sufficient condition
In addition to the necessary condition, the sufficient condition must be met.
At this point, the inflection point is considered independently of its slope. This is covered in detail later.
For the third derivative of a function at the point, the sufficient criterion for an inflection point is:
Another way to determine an inflection point is to look at the second derivative at . If there is a change of sign there, then there is a turning point at this point.
This procedure is usually useful when a diagram is given or the formation of the third derivative is significantly more complex.
Now check the sufficient condition for the input example.
Now look at the whole thing again in mathematical terms.
If for the job
holds, a saddle point exists.
Danger! The saddle point is often wrongly interpreted as an extreme point. However, the saddle point is a turning point and NOT an extreme point.
The best example of a saddle point is the function .
The graph of the function with looks like this:
Figure 2: Graph of a 3rd degree function with a saddle point
It can be seen that the function at has a saddle point.
First form the first, second and third derivative.
Next, apply the necessary criterion.
Now try to apply the sufficient criterion.
In this case, the third derivative is also equal to 0. Here, the possibility of looking at the first derivative at the point comes into play.
This means that all three conditions for a saddle point are met and it is therefore an inflection point, more precisely a saddle point, at the point .
Calculate slope at turning point – right-left turning point
In the case of turning points, a distinction can still be made between a turning point with a positive and a negative slope. A turning point with a positive slope is also called a right-left turning point (RLW) and a turning point with a negative slope is also called a left-right turning point (LRW). The designations come from the fact that a RLW leads from a right turn to a left turn – with an LRW it is the other way around.
To see the difference between an LRW and an RLW, look at a short example.
But what does the LRW and the RLW mean for the sufficient criterion?
For the second derivative of a function at the point, the following is a sufficient criterion for…
… a left-right turning point (LRW) with a positive slope:
… a right-left turning point (RLW) with a negative slope:
Another way to determine an LRW or RLW is to look at the second derivative at . If there is a sign change from + to -, then there is an LRW at this point. If there is a sign change from – to +, then there is an RLW at this point.
This procedure is usually useful when a diagram is given or the formation of the third derivative is significantly more complex.
In this case, you can also look at the introductory example.
A saddle point always has a slope of 0 and thus neither a positive nor a negative slope. In addition, it can change from a left curve to a right curve or vice versa.
Inflection point – calculate x and y value
Since you already know the necessary and sufficient criterion for turning points, you can use the following recipe to calculate turning points.
- Necessary criterion
- after dissolve
- sufficient criterion
- Calculate the y-value of the inflection point
Steps 2 and 3 can be performed multiple times if the function has multiple inflection points.
Calculate turning point – task
In the introductory example, the appropriate y-values for the turning points are still missing.
Great, now you’ve done your homework.