Zeros and a Harmonic Bound

★ 8/10functions Extrema calculus Monotonicity Logarithms

Problem

Let $f(x) = \dfrac{a}{x} + \dfrac12\ln x - 1$ .

1. Discuss the monotonic intervals of $f$ . 2. If $f$ has two zeros $x_1, x_2$ , prove that

\frac{\mathrm{e}(x_1 + x_2)}{x_1x_2} > 2.

Answer

Solution

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

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