IMO 1967 LL BUL3

Origin: BUL

Problem

Prove the trigonometric inequality cos x < 1 −x2 2 + x4 16, where x \in(0, \pi/2).

Solution

Consider the function f : [0, \pi/2] \toR defined by f(x) = 1 −x2/2 + x4/16 −cos x. It is easy to calculate that f ′(0) = f ′′(0) = f ′′′(0) = 0 and f ′′′′(x) = 3/2 −cos x. Since f ′′′′(x) > 0, f ′′′(x) is increasing. Together with f ′′′(0) = 0, this gives f ′′′(x) > 0 for x > 0; hence f ′′(x) is increasing, etc. Continuing in the same way we easily conclude that f(x) > 0.