Consider the function f(x)=x3−3x2+4 defined on the real line.
(i) Explain clearly what is meant by (a) a local extremum and (b) a point of inflection of a differentiable function.
(ii) Show that a point where f′(x)=0 but f′′(x)=0 need not be a local extremum by quoting an explicit example (need not be taken from f above).
(iii) For the given function f(x), identify all x for which f′′(x)=0 and decide, with reasons, whether each such point is a local extremum, a point of inflection, or neither.