Reciprocal Logs Exceed Two

★ 7/10calculus Monotonicity Logarithms inequality

Problem

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

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. If $f(x)$ has two zeros $x_1, x_2$ , prove that

\frac{1}{\ln x_1} + \frac{1}{\ln x_2} > 2.

Answer

Solution

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

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