Concavity, points of inflection and the second derivative test

Learning objectives

Determine the intervals on which a function is concave up or concave down from the sign of f″, and connect concavity to whether f′ is increasing or decreasing

Locate points of inflection where concavity changes, checking that f″ actually changes sign rather than merely equaling zero, and find them from a formula or from a graph of f′

Apply the second derivative test to classify a critical point at which f′ = 0, and recognize when the test is inconclusive so that the first derivative test is needed instead

Connect the graphs of f, f′ and f″: match a function to its derivatives, and read the extrema, inflection points and concavity of f from a graph of f′

Sketch the graph of a function from information about f′ and f″, and sketch a possible graph of f′ or of f from the graph of the other

Justify conclusions about extrema, concavity and inflection points using sign analysis, in the form expected in a free-response solution