What is a turning point?
A turning point is a point where the graph of a function has the locally highest value (called a maximum turning point) or the locally lowest value (called a minimum turning point). A function does not have to have their highest and lowest values in turning points, though.
This graph e.g. has a maximum turning point at (0|-3) while the function has higher values e.g. in (2|5). This implies that a maximum turning point is not the highest value of the function, but just locally the highest, i.e. there is no higher value at least in a small area around that point.
How to find turning points?
The basic idea is that tangents in a turning point have slope
So the basic idea of finding turning points is:
- Find a way to calculate slopes of tangents (possible by differentiation).
- Find when the tangent slope is . There could be a turning point (but there is not necessarily one!)
This means: To find turning points, look for roots of the derivation.
Does slope always imply we have a turning point?
No. If the slope is
, we max have a maximum turning point (shown above)
or a mininum turning point
or the slope just becomes
for a moment though you have no turning point. Such a point is called saddle point.
Does the slope always have to be in turning points?
Yes. This is right. But not the reversion, as seen above. So we say: Having slope
is necessary but not sufficient for having a turning point.
Let's assume we have slope . How can I find out if I have a maximum / minimum turning point or a saddle point?
By using the change of signs criterion.
I have to find turning points as homework and don't know how. What can I do?
Just enter your function above. Mathepower does the calculation, with explanation and step-by-step.