A Two-Variable Uniform Bound

★ 7/10Extrema calculus Completing the Square AM-GM inequality

Problem

Suppose that for all $m, n \in \mathbb{R}$ , the inequality

m - n \leqslant (x - m)^2 + \mathrm{e}^{x-n} - a

holds for every real $x$ . Find the maximum possible value of the real number $a$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, calculus, Completing the Square, AM-GM, inequality

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