A polynomial is defined by p(x)=x3−6x2+9x for all real numbers x. On the closed interval [0,5] the graph of p is continuous, differentiable, and changes concavity at exactly one point.
(a) Compute the average rate of change of p on [0,5].
(b) Solve for all values c in (0,5) that satisfy p′(c) equal to the value found in part (a). Express exact values and give decimal approximations to three decimal places.
(c) The second derivative of p is p′′(x)=6x−12. Identify the x-coordinate of the sole point of inflection of p and determine the intervals on which p is concave up and concave down on (0,5).
(d) For each value of c found in part (b), state whether it lies to the left of, to the right of, or exactly at the point of inflection identified in part (c). Explain how the concavity information supports your conclusion about the relative positioning of the tangent and secant lines involved.