Let g be a twice–differentiable function defined on all real numbers such that g′(2)=0 and g′′(2)=0. You are also told that g′′′(2)=0.
(a) Explain why the usual information provided by g′′(2) is insufficient to determine whether x=2 corresponds to a local extremum of g. Your explanation should reference the definitions of a local maximum and a local minimum without invoking specific theorems by name.
(b) Starting from the definition of a critical number, describe one additional analytic method—other than computing g′′(2)—that could decide whether x=2 is a local maximum, a local minimum, or neither. Provide a justification for why this method succeeds where the information in g′′(2) fails.
(c) Suppose further that g′′′(2)<0. Using the method you described in part (b), determine whether x=2 corresponds to a local maximum, local minimum, or neither for g. Show all reasoning.