Concavity · Andromeda

Definition

Concavity describes the direction in which a curve “bends” or “opens.” It is a measure of the rate of change of the slope (the second derivative).

It allows us to predict whether a trend is accelerating or slowing down, which is critical for optimization and risk assessment.

Concave Up (Convex): $f'$ $f^{'}$ is increasing; $f'' > 0$ $f^{''} > 0$ . The tangent lines lie below the curve.
- How to read: “The second derivative f double prime is greater than zero.”
- Meaning: Slope is rising as $x$ increases—the curve cups upward like a bowl.
Concave Down: $f'$ $f^{'}$ is decreasing; $f'' < 0$ $f^{''} < 0$ . The tangent lines lie above the curve.
- How to read: “The second derivative f double prime is less than zero.”
- Meaning: Slope is falling—the curve bends downward like an arch.
Inflection Point: A point where the concavity changes (where $f''$ changes sign).