Why does the second derivative tell you concavity

Concavity describes how a curve bends relative to its tangent line. If the slope f'(x) grows as x increases, the tangent line tilts upward faster and the graph forms a ∪ shape; this is captured by f''(x)>0. Conversely, if the slope shrinks, the graph bends downward, giving a ∫ shape and f''(x)<0. The second derivative is precisely the derivative of the slope, so its sign records this bending behavior.

How the sign of f'' determines curvature

Geometrically, imagine a tiny segment of the curve and its tangent line. When f''>0, the curve lies above its tangent, creating a cup that can hold water; when f''<0, the curve sits below its tangent, forming a cap. A concrete example is f(x)=$x^3-3x$. Here f'(x)=3x^2-3 and f''(x)=6x. At x=1, f''(1)=6>0, so the graph is concave up near x=1. At x=-1, f''(-1)=-6<0, so it is concave down there. At x=0, f''(0)=0, and the sign changes, indicating an inflection point.

Key points about concavity:

• If f''>0, the curve is concave up (cup‑shaped).
• If f''<0, the curve is concave down (cap‑shaped).
• If f''=0, a sign change may signal an inflection point; otherwise the curvature could remain the same.

Procedure to test concavity of a function:

1. 1Compute the first derivative f'(x).
2. 2Differentiate again to obtain f''(x).
3. 3Determine the sign of f''(x) on the interval of interest.
4. 4If the sign is uniformly positive, conclude concave up; if uniformly negative, conclude concave down.
5. 5If f'' changes sign, locate the points where f''=0; those are candidate inflection points.

Sign of the second derivative versus curve shape: