The polynomial function h(x)=x3−4x is graphed for −3≤x≤3. Its graph intersects the x-axis three times, forming three distinct lobes that are symmetric about the origin.
(a) Find the x-coordinates of all x-intercepts of the graph of h.
(b) For each sub-interval determined by consecutive intercepts, state whether the graph of h lies above or below the x-axis. Justify your answer analytically.
(c) Compute the total area of the regions enclosed between the graph of h and the x-axis on [−3,3]. Show all steps in setting up and evaluating the necessary integral(s).
(d) Explain explicitly how any symmetry of h allowed you to streamline your calculation.