Tangent, Sine, and a Multiplier

★ 7/10trigonometry calculus Monotonicity AM-GM inequality

Problem

Let $0 < x < \dfrac{\pi}{2}$ .

1. Prove that $\tan x + \sin x > 2x$ . 2. Find the largest positive integer $n$ such that

\tan x - x > n\,(x - \sin x)

holds for all such $x$ .

Answer

Solution

Difficulty	7/10
Topics	trigonometry, calculus, Monotonicity, AM-GM, inequality

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