A Critical Constant Above 1.3

★ 9/10Extrema calculus Convexity Monotonicity inequality

Problem

Let $f(x) = \dfrac{e^x - 1}{x}$ .

1. Find the intervals of monotonicity of $f(x)$ . 2. Suppose

e^x - 2x\ln x - kx - 1 \geqslant 0 \qquad \text{for all } x > 0,

and let $\lambda$ be the maximum such $k$ . Prove that $\lambda > 1.3$ .

Answer

Solution

Difficulty	9/10
Topics	Extrema, calculus, Convexity, Monotonicity, inequality

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