A twice–differentiable function f is defined on all real numbers. At x=c the function satisfies f′(c)=0 and f′′(c)=0.
(a) Define what it means for a real number x=c to be a critical number of f. Your answer should reference both the existence and value of the derivative.
(b) Explain, without using any specific function, why f′′(c)>0 guarantees that f attains a local minimum at x=c, whereas f′′(c)<0 guarantees a local maximum. Your explanation should explicitly connect concavity to the shape of the graph near x=c.
(c) Construct an explicit function g for which x=0 is a critical number but g′′(0)=0, making the second–derivative test inconclusive. Justify why the test fails and determine, using another method, whether x=0 is a local extremum for your g. Clearly state your conclusion.