Use Exercise 59 and the inequality sin x ≤ x for x ≥ 0 (established in Theorem 3 of Section 2.6) to prove the following assertions for all x ≥ 0 (each assertion follows from the previous one).
(a) cos x ≥ 1 - 1/2x2
(b) sin x ≥ x - 1/6x3
(c) cos x ≤ 1 - 1/2x2 + 1/24 x4
(d) Can you guess the next inequality in the series?
Exercise 59
Prove that if f (0) = g(0) and f′(x) ≤ g′(x) for x ≥ 0, then f (x) ≤ g(x) for all x ≥ 0.