# Convexity and Concavity

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Convexity and Concavity

a property of the graph of the function y = f(x) (a curve) that consists in the fact that each arc of the curve lies neither higher nor lower than its chord. In the first instance the graph of the function f(x) is convex downward (concave upward) and the function itself is called convex; in the second instance the graph is concave downward (convex upward) and the function is called concave. If there are derivatives f′(x) and f′(x), then the first instance applies when f′(x)= >̳ 0 and the second when f′(x) <̳ 0 (at all points along the interval under consideration). Convexity downward can also be characterized by the fact that the arc of the curve lies no lower than the tangent at any of its points, and concavity downward by the fact that the arc of the curve lies no higher than the tangent. Convexity and concavity of a surface are defined analogously.