Concavity

Concavity describes the curvature of a real-valued function. In one variable, a function f defined on an interval I is concave if for all x, y in I and t in [0, 1], f(t x + (1 - t) y) ≥ t f(x) + (1 - t) f(y). The graph lies above the chord between any two points. If the inequality is strict for x ≠ y, f is strictly concave. If f is twice differentiable, concavity is equivalent to f''(x) ≤ 0 on I; convexity corresponds to f''(x) ≥ 0, with the inequality directions reversed in the definitions.

For differentiable functions, concavity also means the graph lies below its tangent lines: f(y) ≤ f(x) + f'(x)(y - x) for all x, y in I.

In several variables, a function f: D ⊆ R^n → R is concave if D is convex and f(t x + (1 - t) y) ≥ t f(x) + (1 - t) f(y) for all x, y in D and t in [0, 1]. If f is twice differentiable, this is equivalent to the Hessian matrix H_f(x) being negative semidefinite for all x in D.

Inflection points mark changes in concavity, often where f'' changes sign; their existence depends on smoothness and the function’s form.

Concavity has various applications: in optimization, a concave function on a convex domain attains a global maximum at any local maximum; in economics, concavity models diminishing marginal utility; and in probability, log-concave distributions have useful properties.