Consider the polynomial function p(x)=x3−6x2+9x+1 on the interval [0,3].
(a) Determine the average slope of p over [0,3].
(b) Solve p′(x)=m, where m is the value obtained in part (a), and list all solutions.
(c) Verify that every solution from part (b) lies inside the open interval (0,3) and interpret what each solution represents in the context of the graph of y=p(x).