Concavity
Concavity describes which way a curve bends: a function is concave up where its graph curves like a cup and concave down where it curves like a frown. It is determined by the sign of the second derivative.
Concavity captures the direction a graph bends, independent of whether it is rising or falling. A function is concave up on an interval where its graph opens upward (holds water) and concave down where it opens downward. Equivalently, a curve is concave up where it lies above its tangent lines and its slope is increasing, and concave down where it lies below its tangent lines and its slope is decreasing.
The second derivative is the standard test: where f″(x) > 0, the function is concave up; where f″(x) < 0, it is concave down. For example, f(x) = x³ has f″(x) = 6x, so the curve is concave down for x < 0 and concave up for x > 0. A point where concavity changes — and the function is continuous there — is an inflection point; for x³ that happens at the origin.
Concavity also powers the second derivative test for classifying critical points. If f′(c) = 0 and f″(c) > 0, the graph is concave up at c, so f has a local minimum there; if f″(c) < 0, it is concave down and f has a local maximum. Beyond curve sketching, concavity carries meaning in applications: on a position graph, concave up means acceleration is positive, and in economics, concavity distinguishes increasing from diminishing rates of change.
AP Calculus AB tests concavity directly — finding intervals of concavity, locating inflection points, and applying the second derivative test — and the FE Civil and FE Mechanical exams include it within their differential calculus coverage.
Key takeaways
- Concave up means the graph bends upward and lies above its tangent lines; concave down means the opposite.
- f″(x) > 0 implies concave up; f″(x) < 0 implies concave down.
- An inflection point is where the graph changes concavity.
- The second derivative test uses concavity to classify critical points as local minima (f″ > 0) or maxima (f″ < 0).
- AP Calculus AB and the FE exams test concavity intervals, inflection points, and the second derivative test.