The Largest $a$ on an Interval

★ 7/10Extrema calculus Monotonicity Logarithms inequality

Problem

If the inequality

\mathrm{e}^x - a\ln(ax - 1) + 1 \geqslant 0

holds for every $x \in \left[\dfrac{1}{2}, 1\right]$ (where $\mathrm{e}$ is the base of the natural logarithm), find the maximum possible value of the real number $a$ .

Answer

Solution

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

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