المجموعات والدوالّ المُقعّرة والمحدبة ... Convex, Concave Sets and Functions

يشرح هذا المقطع مفهوم المجموعة المُقعّرة والمجموعة المُحدبة والدالة المُقعّرة والدالة المُحدبة والمعنى الاقتصادي لكل منها مدعوماً بالتوضيح البياني ... Convex, Concave Sets & Functions

In #mathematics, a set or a function is said to be convex or concave depending on the direction in which the curve of the set or function is bent. A set or function is said to be convex if a line joining any two points on the curve lies completely within the curve. On the other hand, a set or function is said to be concave if the curve bends downwards, and the line joining any two points on the curve lies above the curve.

In terms of functions, a function is said to be convex if its second derivative is positive or non-negative, meaning that the curve of the function bends upwards or stays flat. Conversely, a function is said to be concave if its second derivative is negative or non-positive, meaning that the curve of the function bends downwards or stays flat.

The concepts of #convexity and #concavity are important in optimization and economics, as they are used to analyze the properties of functions and sets, and to determine optimal solutions to problems. For example, a convex optimization problem is one in which the objective function is convex, while a concave optimization problem is one in which the objective function is concave. The shape of the function or set can have a significant impact on the optimization solution, making convexity and concavity important concepts to consider in these fields.