Concavity

Concavity describes the way a curve bends — whether it curves upward like a bowl or downward like a hill. When a function is concave up on an interval, its graph bends upward and any tangent line drawn to the curve lies below the graph. When it is concave down, the graph bends downward and tangent lines lie above it. Understanding concavity gives us a deeper picture of a function's behavior beyond just whether it is increasing or decreasing. The second derivative of a function, f''(x), is the primary tool for determining concavity. Because the second derivative measures how the first derivative is changing, a positive f''(x) means the slope is increasing — which produces an upward-curving shape — while a negative f''(x) means the slope is decreasing, producing a downward-curving shape. This connection transforms concavity from a visual idea into something we can calculate precisely. Concavity analysis is essential in curve sketching, optimization, and understanding real-world phenomena such as acceleration, the diminishing returns of a production function in economics, or the bending of a beam in engineering. Together with information about critical points, concavity lets us classify local maxima and minima using the Second Derivative Test, making it one of the most powerful tools in differential calculus.

Key takeaways
  • Find the Second Derivative
  • Find Potential Inflection Points
  • Build a Sign Chart for f''(x)
  • Confirm Inflection Points
  • Apply the Second Derivative Test (Optional)

Find the Second Derivative

Start by differentiating f(x) once to obtain f'(x), then differentiate again to obtain f''(x). For example, if f(x) = x^4 - 4x^3, then f'(x) = 4x^3 - 12x^2 and f''(x) = 12x^2 - 24x. This second derivative is the function whose sign controls concavity.

Find Potential Inflection Points

Set f''(x) = 0 and solve. Also check where f''(x) is undefined. These x-values are candidates for inflection points — places where concavity might change. For 12x^2 - 24x = 0, factor to get 12x(x - 2) = 0, giving x = 0 and x = 2 as candidates.

Build a Sign Chart for f''(x)

The candidate x-values divide the real number line into intervals. Choose a test value inside each interval and evaluate f''(x) at that test value. If f''(x) > 0 on the interval, the function is concave up there. If f''(x) < 0, the function is concave down. For the example: on (-∞, 0) test x = -1: f''(-1) = 12 + 24 = 36 > 0, so concave up. On (0, 2) test x = 1: f''(1) = 12 - 24 = -12 < 0, so concave down. On (2, ∞) test x = 3: f''(3) = 108 - 72 = 36 > 0, so concave up.

Confirm Inflection Points

An inflection point occurs at x = c only if f''(c) = 0 (or is undefined) AND the sign of f''(x) actually changes across x = c. If the sign does not change, there is no inflection point even though f''(c) = 0. For our example both x = 0 and x = 2 show a sign change, confirming inflection points at (0, f(0)) = (0, 0) and (2, f(2)) = (2, -16).

Apply the Second Derivative Test (Optional)

At a critical point c where f'(c) = 0: if f''(c) > 0 the function has a local minimum at c; if f''(c) < 0 it has a local maximum at c; if f''(c) = 0 the test is inconclusive and you must use another method such as the First Derivative Test.

The Key Equation

f′′(x)>0  ⟹  concave up;f′′(x)<0  ⟹  concave downf''(x) > 0 \implies \text{concave up}; \quad f''(x) < 0 \implies \text{concave down}
  • f(x) — Original function (varies): The function whose graph we are analyzing for concavity.
  • f'(x) — First derivative (varies): Measures the instantaneous rate of change (slope) of f at each x.
  • f''(x) — Second derivative (varies): Measures how the slope f'(x) is changing; determines concavity of the graph.

Worked Example

Problem. Let f(x) = x^3 - 6x^2 + 9x + 1. Find the intervals of concavity and any inflection points.

Concave down on (−∞,2);  concave up on (2,∞);  inflection point (2,3)\text{Concave down on } (-\infty, 2);\; \text{concave up on } (2, \infty);\; \text{inflection point } (2, 3)

Answer. Concave down on (-∞, 2), concave up on (2, ∞), inflection point at (2, 3). Local max at (1, 5); local min at (3, 1).

Figure
Diagram
Concavity: f(x) = x³ − 6x² + 9x + 1xfLocal max (1, 5)Local min (3, 1)Inflection (2, 3)Concave ↓Concave ↑f''<0f''>0123
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus alongside Isaac Newton and introduced much of the notation still used today, including dy/dx for derivatives and the elongated-S symbol ∫ for integrals. His systematic notation made it natural to consider higher-order derivatives such as d²y/dx², which are the foundation of concavity analysis. Leibniz also explored the geometric meaning of the second derivative in the context of curve analysis. Leibniz designed one of the earliest mechanical calculators, the Stepped Reckoner, which could multiply and divide — a remarkable feat of engineering in the 17th century.

  • First Derivative Test — The First Derivative Test uses the sign of f'(x) to classify local extrema; concavity analysis via the second derivative provides an alternative classification method.
  • Inflection Points — Inflection points are precisely the x-values where concavity changes, identified by finding where f''(x) = 0 and confirming a sign change.
  • Curve Sketching — Complete curve sketching combines information from f, f', and f'' — including concavity — to produce an accurate graph of a function.
  • Second Derivative Test — The Second Derivative Test directly uses concavity (the sign of f'' at a critical point) to determine whether that point is a local max or min.
  • Higher-Order Derivatives — The second derivative is a higher-order derivative; understanding how to compute and interpret these is prerequisite knowledge for all concavity work.