Concavity, points of inflection, curve sketching and optimization

Learning objectives

Determine concavity from the sign of f″ and connect it to whether the first derivative is increasing or decreasing

Locate points of inflection where f″ changes sign, and explain why f″ = 0 on its own does not guarantee an inflection point

Apply the second derivative test at a point where f′ = 0, and revert to the first derivative test when the second derivative is zero or undefined there

Move between the graphs of f, f′ and f″, matching each to the others and sketching any one of them from information about the rest

Solve an optimization problem by building an objective function of a single variable from a constraint, stating the domain, and justifying that the extremum found is absolute rather than merely local

Analyze an implicitly defined curve for horizontal and vertical tangents and for concavity, using dy/dx and the second derivative obtained by differentiating it again implicitly