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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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).