A Paper Tiger

★ 8/10Extrema calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = (2-x)\mathrm{e}^{k(x-1)} - x$ with $k$ real.

1. If $f$ is decreasing on all of $\mathbb{R}$ , find the maximum value of $k$ . 2. For $x \in (1, 2)$ , prove that

\ln\frac{x(2x-1)}{2-x} > 2\left(x - \frac1x\right).

Answer

Solution

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

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