Concavity
Concavity describes whether a curve bends upward like a bowl (concave up) or downward like a dome (concave down), determined by the sign of the second derivative.
Formula
f''(x) > 0: \text{ concave up; } f''(x) < 0: \text{ concave down}
Definition
Concavity describes which way a curve bends: concave up means the curve looks like a smile or bowl, concave down means it looks like a frown or dome. Formally, $f$ is concave up on an interval if $f'' > 0$ there (tangent lines lie below the curve) and concave down if $f'' < 0$ (tangent lines lie above); concavity changes at inflection points. More generally, a function is convex (concave up) if for all $x, y$ in the interval and $t$ in $[0,1]$: $f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$, equivalent to $f'' \ge 0$ when $f$ is twice differentiable.
Example
$y = x^2$ is concave up everywhere (opens upward like a U), while $y = -x^2$ is concave down (opens downward like an upside-down U). For $f(x) = x^3 - 3x$: $f''(x) = 6x$ is negative for $x < 0$ (concave down) and positive for $x > 0$ (concave up), with an inflection at $x=0$. Jensen's inequality follows from convexity: for a convex function $f$ and random variable $X$, $E[f(X)] \ge f(E[X])$, underlying many statistical bounds.
Key Insight
A cup holds water if concave up, a dome sheds water if concave down, an easy way to remember which is which, and knowing concavity improves graph sketching dramatically since it tells you which way the curve bends between critical points. Convexity ensures that any local minimum is a global minimum, a property exploited extensively in machine learning and operations research.