Let f(x)=x4−8x3+18x2+1, defined for all real numbers.
(a) Calculate f′′(x). Show all work.
(b) Find all real values of x at which f′′(x)=0 or f′′(x) is undefined. These are the possible points of inflection.
(c) For each interval determined by the values found in part (b), determine the sign of f′′(x). State the resulting intervals of concavity for f.
(d) Identify all points of inflection of the graph of f, if any, and justify your answer.