CONCAVE FUNCTION Meaning and
Definition
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A concave function is a mathematical function that exhibits a downward curvature on its graph. More formally, a function f(x) defined on an interval is said to be concave if, for any two points (x1, y1) and (x2, y2) on the graph of the function, the line segment joining these two points lies entirely below or on the graph of the function.
In simpler terms, a concave function can be visualized as a function that curves inward like a bowl. The graph of a concave function slopes downwards, implying that as the input values increase, the rate of change of the function decreases. This characteristic distinguishes it from a convex function, which curves outward and has an upward slope.
A concave function possesses several important properties. Firstly, it guarantees that any local maximum point is also the global maximum point. Additionally, the secant lines connecting any two points on the graph will always lie below the graph itself. This property is known as the Jensen's inequality and has various applications in mathematics and economics.
Common examples of concave functions include the square root function, f(x) = √x, and the logarithmic function, f(x) = log(x). These functions exhibit the characteristic concave curvature and their graphs are concave downwards.
Etymology of CONCAVE FUNCTION
The word concave originates from the Latin term concavus, which means hollow or arched inward. It is derived from the verb cavare, which means to make hollow or excavation. In the context of mathematics, a concave function describes a function that is curved or arched inward in its graph. The term describes the shape of the graph, mimicking a hollow or concave shape, which led to the adoption of the term concave function.
