A Weighted Log Inequality

★ 7/10Extrema calculus Substitution Monotonicity inequality

Problem

The function $f(x) = ax(\ln x - 1) - x^2$ (with $a \in \mathbb{R}$ ) has exactly two extreme points $x_1 < x_2$ .

1. Find the range of possible values of $a$ . 2. If the inequality

\ln x_1 + \lambda\ln x_2 > 1 + \lambda

always holds, find the range of possible values of the real number $\lambda$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, calculus, Substitution, Monotonicity, inequality

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