Concavity
Concavity describes the curvature direction of a function's graph — a function is concave up on an interval if its graph curves upward like a bowl, and concave down if it curves downward like a dome. The second derivative determines concavity: positive second derivative implies concave up; negative implies concave down. Understanding concavity helps identify local extrema and inflection points in curve sketching and optimization problems.
Key Formula
f''(x) > 0 means concave up; f''(x) < 0 means concave down
LaTeX: f''(x) > 0 \Rightarrow \text{concave up}; \quad f''(x) < 0 \Rightarrow \text{concave down}
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Function of x | dimensionless |
| f''(x) | Second derivative | dimensionless |
Worked Example
Problem
Determine the intervals of concavity for f(x) = x⁴ − 4x³.
Solution
Step 1: First derivative: f'(x) = 4x³ − 12x². Step 2: Second derivative: f''(x) = 12x² − 24x = 12x(x − 2). Step 3: Set f''(x) = 0: 12x(x − 2) = 0 → x = 0 and x = 2. Step 4: Test intervals: x < 0: f''(−1) = 12(−1)(−3) = 36 > 0 → concave up. 0 < x < 2: f''(1) = 12(1)(−1) = −12 < 0 → concave down. x > 2: f''(3) = 12(3)(1) = 36 > 0 → concave up.
Answer
Concave up on (−∞, 0) and (2, ∞); concave down on (0, 2); inflection points at x = 0 and x = 2
Concavity Summary Table
| Sign of f''(x) | Concavity | Graph Shape | Tangent Lines | Example Function |
|---|---|---|---|---|
| f''(x) > 0 | Concave Up | Bowl (∪) | Below the curve | f(x) = x² |
| f''(x) < 0 | Concave Down | Dome (∩) | Above the curve | f(x) = −x² |
| f''(x) = 0 (no sign change) | Neither | Flat curvature | On the curve | f(x) = x⁴ at x=0 |
| f''(x) = 0 (sign change) | Inflection | Transition | Crosses curve | f(x) = x³ at x=0 |
Interactive Tools
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Related Terms
Inflection Point
An inflection point is a point on a curve where the concavity changes — the curve transitions from concave up to concave down, or vice versa. At such a point, the second derivative of the function is zero or undefined. Inflection points are critical in optimization, economics, and physics for identifying transitions in the rate of change of a rate of change.
Critical Point
The critical point is the endpoint of the liquid–gas phase boundary on a phase diagram, defined by the critical temperature (T_c) and critical pressure (P_c), beyond which the distinction between liquid and gas phases disappears and the substance exists as a supercritical fluid. Above the critical temperature, no amount of pressure can liquefy the gas; above the critical pressure at the critical temperature, the fluid exhibits properties intermediate between gas and liquid. Supercritical fluids have important industrial applications, such as supercritical CO₂ in extraction processes.
From Latin "concavus" meaning "hollow" or "vaulted." The mathematical usage of concavity developed alongside second-derivative analysis in the 18th and 19th centuries.