AHL 5.10 (HL)—Second derivative and concavity

Concavity describes whether a curve bends above or below its tangent trend. The sign of the second derivative gives this local curvature when the function is twice differentiable.

An inflection point is where concavity changes, not merely where f''=0. Check the sign on both sides and include points where f'' is undefined if the original function is defined there.

For f(x)=x³, f''(x)=6x changes from negative to positive at x=0, so the origin is a stationary inflection: f'(0)=0 but the function keeps increasing through it.

A zero second derivative alone does not prove an inflection or extremum. Use neighbouring intervals and the domain, not one substituted value.