A Sum Bounded Below

★ 7/10calculus Convexity Monotonicity Logarithms inequality

Problem

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

1. Discuss the monotonicity of $f(x)$ . 2. If $x_1, x_2$ are two real solutions of the equation $f(x) = 2$ , prove that

x_1 + x_2 > \frac{2}{\mathrm{e}^3}.

Answer

Solution

Difficulty	7/10
Topics	calculus, Convexity, Monotonicity, Logarithms, inequality

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