A Positive Gap

★ 7/10Extrema calculus Monotonicity inequality

Problem

Let $f(x) = \mathrm{e}^x\left(x\ln x + \dfrac{2}{\mathrm{e}}\right)$ .

1. Find the intervals on which $h(x) = f(x) - \mathrm{e}^x\left(1 + \dfrac{2}{\mathrm{e}}\right)$ is monotonic. 2. Prove that $f(x) - x > 0$ .

Answer

Solution

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

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