A concave function is a type of mathematical function where a line segment connecting any two points on the graph of the function lies below or on the graph itself. This means that the function curves downwards, resembling a bowl turned upside down. In mathematical terms, a function f(x) is concave if, for any two points x_1 and x_2 , the inequality f(tx_1 + (1-t)x_2) \geq tf(x_1) + (1-t)f(x_2) holds for all t in the interval [0, 1]. Concave functions are important in various fields, including economics and optimization. For example, the utility function in economics often exhibits concavity, indicating diminishing returns. Additionally, calculus provides tools to analyze concavity through the second derivative test, where a negative second derivative indicates that a function is concave. Understanding conc