A Root-Exponent Comparison

★ 8/10calculus Monotonicity Logarithms inequality

Problem

For all reals $m, n$ with $0 < n < m < \dfrac{1}{a}$ , the inequality

\frac{\sqrt[m]{n}}{\sqrt[n]{m}} > \frac{n^a}{m^a}

always holds. Find the minimum value of the real number $a$ .

Answer

Solution

Difficulty	8/10
Topics	calculus, Monotonicity, Logarithms, inequality

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