Concavity
Calculus & Advanced MathConcavity 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.
Example
$y = x^2$ is concave up everywhere (it opens upward like a U). $y = -x^2$ is concave down (it opens downward like an upside-down U).
Key Insight
A cup holds water if concave up. A dome sheds water if concave down. That is the easiest way to remember which is which.
Definition
$f$ is concave up on an interval if $f'' > 0$ there (tangent lines lie below the curve). $f$ is concave down if $f'' < 0$ (tangent lines lie above). Concavity changes at inflection points.
Example
$f(x) = x^3 - 3x$: $f''(x) = 6x$. Concave down for $x < 0$ ($f'' < 0$), concave up for $x > 0$ ($f'' > 0$). Inflection at $x = 0$.
Key Insight
Knowing concavity improves graph sketching dramatically: it tells you which way the curve bends between critical points.
Definition
A function $f$ is convex (concave up) on an interval if for all $x, y$ in the interval and $t$ in $[0,1]$: $f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$. This is the global analytic definition, equivalent to $f'' \ge 0$ when $f$ is twice differentiable. Convexity is a central concept in optimization theory.
Example
Jensen's inequality: for a convex function $f$ and a random variable $X$, $E[f(X)] \ge f(E[X])$. This underlies information-theoretic inequalities and many statistical bounds.
Key Insight
Convexity (concave up) ensures that any local minimum is a global minimum, a property exploited extensively in machine learning and operations research.