(a) Graph several members of the family of functions f(x) = (2cx – x2)/c3 for c > 0 and look at the regions enclosed by these curves and the -axis. Make a conjecture about how the areas of these regions are related.
(b) Prove your conjecture in part (a).
(c) Take another look at the graphs in part (a) and use them to sketch the curve traced out by the vertices (highest points) of the family of functions. Can you guess what kind of curve this is?
(d) Find the equation of the curve you sketched in part (c).