An antiderivative f of the function f′(x)=x3−6x2+9x is defined for all real x.
(a) Compute f′′(x). Factor completely.
(b) Determine the x–values at which f′ has a local extremum. Show analytical work that leads to your conclusion.
(c) Decide all points of inflection of f. For each point, verify explicitly that the concavity of f changes sign.
(d) State the intervals on which f is concave up and the intervals on which it is concave down.