What is the use of the change of sign?By checking for the change of sign, you can check whether a function with derivative has a maximum / minimum turning point or a saddle point. Here are three examples where the function has slope in (1|2):
This function has slope in (1|2) and a maximum turning point. At the graph ascends, i.e. the derivative is larger than in here. At the Graph falls, i.e. the derivative is less than . This means that around a maximum turning point, the sign of the derivative is + before and - after the turning point. This means the derivative changes signs from + to - .
This function has slope in (1|2), but a minimum turning point. At the graph falls, i.e. the derivation here is less than . At the graph ascends, i.e. the derivation is larger than . This implies that at a minimum turning point, the sign of the derivation is - before and + after the turning point. This means the derivation changes signes from - to +.
This function also has slope at (1|2), but no turning point. You see that the graph ascends at as well as at . This implies you have no turning point if the derivation does not change signs. Such a point (that is no turning point but has derivation ) is called saddle point.
How to use the change of sign criterion?
- First derivate your function.
- Then calculate the roots of the derivation. Only those roots can be x-coordinates of turning points.
- Then you insert x-values close to the derivation roots into the derivation. If the derivation changes signs around the derivation, you found a turning point. Otherwise not.
Why is change of signs called a sufficient criterium?The derivation being is necessary for a turning point (i.e. it is always the case at a turning point). But it is not necessary, what means, just because the derivation is , there does not have to be a turning point (check out for the saddle point above).
If the derivation is not only , but also changes signs, then you have to have a turning point. In mathematics, we say that derivation and change of signs is sufficient for having a turning point.