Calculate turning points

Enter the function whose turning points you want to calculate.
Hint: Enter as 3*x^2 ,
as 3/5 and
as (x+1)/(x-2x^4)

To write powers, use ^. This means, you gotta write x^2 for .

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 0.

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 0. 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 0 always imply we have a turning point?

No. If the slope is 0, we max have a maximum turning point (shown above)

or a mininum turning point

or the slope just becomes 0 for a moment though you have no turning point. Such a point is called saddle point.

Does the slope always have to be 0 in turning points?

Yes. This is right. But not the reversion, as seen above. So we say: Having slope 0 is necessary but not sufficient for having a turning point.

Let's assume we have slope 0. 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.