A Trig Bound with a Parameter

★ 7/10Extrema calculus Monotonicity AM-GM inequality

Problem

Let $f(x) = \dfrac{a}{2}\ln(x+1) - \sqrt{x+2}$ , where $a \in \mathbb{R}$ .

1. When $a = \dfrac{8}{3}$ , find the intervals on which $f(x)$ is monotonic. 2. If for all $x \geqslant 0$ ,

f(x) \leqslant \frac{3}{a}(\sin x + \cos x),

find the range of possible values of the real number $a$ .

Answer

Solution

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

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