Prove the Generalized Rolle's Theorem, Theorem 1.10, by verifying the following.
a. Use Rolle's Theorem to show that f'(zi) = 0 for n − 1 numbers in [a, b] with a < z1 < z2 < · · · < zn−1 < b.
b. Use Rolle's Theorem to show that f''(wi) = 0 for n−2 numbers in [a, b] with z1 < w1 < z2 < w2 · · ·wn−2 < zn−1 < b.
c. Continue the arguments in a. and b. to show that for each j = 1, 2. . . n − 1 there are n - j distinct numbers in [a, b] where f(j) is 0.
d. Show that part c. implies the conclusion of the theorem.