Three Zeros of an Absolute Log

★ 8/10functions Log Mean Inequality calculus Absolute Value Monotonicity

Problem

Let $a \neq 0$ be real and

f(x) = |\ln x| + \frac{a}{x}.

1. If $a = 1$ , find the minimum value of $f(x)$ . 2. Discuss the monotonicity of $f(x)$ . 3. If $f(x)$ has exactly three distinct zeros $x_1, x_2, x_3$ , prove that

x_1 + x_2 + x_3 > \frac{2a^2 - 4a + 1}{1 - a}.

Answer

Solution

Difficulty	8/10
Topics	functions, Log Mean Inequality, calculus, Absolute Value, Monotonicity

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