Are turning points and stationary points the same?

Question

I'm currently doing AS maths and my Pure 1 textbook treats stationary points and turning points as the same thing. Furthermore, my teacher said that stationary and turning points are the same. However, from my understanding, a turning point is where the gradient changes sign and a stationary point is where the derivative is 0. Hence they are slightly different, in the sense that a point of inflexion should not be a turning point. I'm slightly confused. Who is right? Could your provide citations and things to read to prove your answer?

Answer 1

from my understanding, a turning point is where the gradient changes sign and a stationary point is where the derivative is 0.

This is exactly right.

a point of inflexion should not be a turning point.

Indeed, inflexion points and turning points are disjoint sets.

I'm currently doing AS maths and my Pure 1 textbook treats stationary points and turning points as the same thing.

No, they are not synonyms:

• $y=|x|$ contains a non-stationary turning point.
• Every point of $y=0$ is a non-inflexion non-turning stationary point.

You didn't ask, but:

• $y=x^3+x$ contains a non-stationary inflexion point.

Page 18 of your syllabus says, "Knowledge of points of inflexion is not included." This is likely the main reason that your textbook is acting as if inflexion points don't exist. My 2nd bullet point above is partly tongue-in-cheek: the exam will not require you (or even expect) to identify those points as stationary points.

Answer 2

I have never heard of these terms, but the definitions are not the same. The derivative changes sign when the derivative becomes zero at a turning point. So if the derivative was positive then becomes $0$, it will be negative after that. But the same doesn’t have to be true for a stationary point, as the comment says below. As @jcneek said, don't let this confuse you and stay with what the course says.

These points are more well known as an extremum (plural is extrema) by the way.