A Telescoping Log Bound

★ 6/10Extrema calculus Monotonicity Telescoping inequality

Problem

Let $f(x) = x^2 + (a-2)x - a\ln x$ with $a \in \mathbb{R}$ .

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

\frac{1}{2^2} + \frac{2}{3^2} + \frac{3}{4^2} + \cdots + \frac{n}{(n+1)^2} < \ln(n+1).

Answer

Solution

Difficulty	6/10
Topics	Extrema, calculus, Monotonicity, Telescoping, inequality

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