(a) The cubic function f(x) = x(x – 2)(x – 6) has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice?
(b) Suppose the cubic function f(x) = (x – a) (x – b) (x – c) has three distinct zeros: a, b, and c. Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros a and b intersects the graph of f at the third zero.