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. Concave functions are important in various fields, including economics and optimization, as they often represent diminishing returns. In mathematical terms, a function f(x) is concave if, for any two points x_1 and x_2 in its domain, the following inequality holds: f(\lambda x_1 + (1 - \lambda)x_2) \geq \lambda f(x_1) + (1 - \lambda)f(x_2) for all \lambda in the interval [0, 1]. Examples of concave functions include quadratic functions with a negative leading coefficient and logarithmic functions.