A Logarithmic Telescope

★ 8/10Extrema calculus Monotonicity Telescoping inequality

Problem

Let $a > 0$ and

f(x) = \ln(1 + ax) - \frac{2x}{x + 2}.

1. When $a = \dfrac{1}{2}$ , find the extreme values of $f(x)$ . 2. Discuss the monotonicity of $f(x)$ on $(0, +\infty)$ . 3. For $n \in \mathbb{N}^*$ , prove that

\frac{2}{2^2 + 3} + \frac{2}{2^3 + 3} + \cdots + \frac{2}{2^{n+1} + 3} \leqslant \frac{2}{7} + \ln\frac{3\cdot 2^{n-1}}{2^n + 1}.

Answer

Solution

Difficulty	8/10
Topics	Extrema, calculus, Monotonicity, Telescoping, inequality

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