A Hidden-Zero Comparison

★ 6/10functions Extrema calculus Monotonicity Casework

Problem

Let $f(x) = \dfrac{a\ln x + 1}{x}$ with $a \in \mathbb{R}$ , and $g(x) = \mathrm{e}^x - 1$ .

1. Discuss the monotonicity of $f(x)$ . 2. If $f(\mathrm{e}) = \dfrac{2}{\mathrm{e}}$ , prove that $g(x) \geqslant f(x)$ .

Answer

Solution

Difficulty	6/10
Topics	functions, Extrema, calculus, Monotonicity, Casework

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