A Golden Logarithmic Inequality

★ 8/10Extrema calculus Logarithms inequality Symmetry

Problem

Prove that for every nonzero real number $x$ ,

\max\{0, \ln|x|\} \geqslant \frac{\sqrt{5}-1}{2\sqrt{5}}\ln|x| + \frac{1}{2\sqrt{5}}\ln\left|x^2-1\right| + \frac{1}{2}\ln\frac{\sqrt{5}+1}{2},

with equality if and only if $x = \pm\dfrac{\sqrt{5}+1}{2}$ or $x = \pm\dfrac{\sqrt{5}-1}{2}$ .

Answer

Solution

Difficulty	8/10
Topics	Extrema, calculus, Logarithms, inequality, Symmetry

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