# Is 'vertex' the correct term for peak (and valley) point on a curve?

This is about mathematical terminology in English.

A simple curving line can have a top peak and a bottom valley. What is the correct term to use for the top/bottom point?

In my native tongue, we call it a `toppunkt`, literally meaning `top-most point` (although it counts for both peaks and valleys). When looking it up I have found the word `vertex`, which Wikipedia agrees with. Is this really the correct term?

Oxford Dictionaries doesn't seem to agree but only defines `vertex` as opposite angular points in a geometrical shape.

`Vertex` literally means `turning point` in Latin, so it does make sense. But from simple google searches, the word `vertex` doesn't seem to be used very often and the term `turning point` actually seems to be used as well.

I would appreciate help with sorting out the terminology here.

Note that I am aware that such a point is called a `stationary point` in 2D (and higher dimensions). I am specifically asking to the 1D case, though, where there is a special term.

• This is very confusing. What do you mean by "a simple curve"? What branch of mathematics or what application of it? Graphs? Celestial navigation? Cartesian geometry? Topology? May 12, 2020 at 16:50

This is quite technical mathematical language, the vertex, or apex is the point at which the curvature is a minimum or a maximum. For a vertical parbola, the vertex occurs at the maximum point. For a car driver, the apex of a turn is the point at which the steering wheel is turned the most.

However, it is more common to speak of

• a "stationary point" (zero gradient, applies to 1D as well higher dimensions),
• a "turning point" (a stationary point that isn't an inflexion) or
• a "(local) maximum/minimum" (which will be at a stationary point, if the curve is differentiable)

"Turning point" is the most useful term for the top or bottom of a curve.

Find the turning points of the curve y = x³ - x, and for each point, determine if it is a maximum or minimum.

• Thank you for the answer. So, you would advise me to use the term `turning point` in technical work rather than `vertex` or `apex`? Or is it a "colloquial" term? Regarding your first point, I am aware of the term `stationary point` which my native language also uses - but of some reason we never use that term in the 1-dimensional case although it strictly is correct. Not even in technical literature. Instead we have the term `toppunkt` to use for 1D. I am assuming that this is also the case in English - but please correct me if wrong - and I am thus looking for the translation of `toppunkt`. May 14, 2019 at 8:21
• If you're talking about a sine wave or similar the usual terms would be "peak" and "trough." Turning point is used as a technical term, along with (local) maxima and (local) minima. May 14, 2019 at 15:29

In mathematics, the vertex is a point at which two lines meet and form an angle. That could refer to the highest peak of a triangle, but it could equally refer to any of the other two angles. The names for any part of a geometric shape usually remain the same regardless of its orientation.

The name for the highest point or top of a shape or object is the apex.

The bottom, or lowest point of a shape is normally just called the base.

I would simply use the words 'Maximum' and 'Minimum'. Sample sentences referring to the x coordinate, y coordinate or both at once are given below.

The curve f(x) has an absolute maximum at (x1,y1) and a local minimum at (x2,y2). The maximum value y1 is reached when x=x1. f'(x)=0 at the point (x2,y2).