A Root-Two Bound

★ 8/10calculus Monotonicity Logarithms inequality Symmetry

Problem

1. Find the maximum of $f(x) = x\ln x - (1-x)\ln(1-x)$ on $0 < x \leqslant \dfrac{1}{2}$ . 2. For $0 < x < 1$ , prove that

x^{1-x} + (1-x)^x \leqslant \sqrt{2}.

Answer

Solution

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

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