A Minimum of Negative One

★ 6/10Extrema calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = a\mathrm{e}^x - \mathrm{e}x - a$ with $a < \mathrm{e}$ .

1. If the local minimum value of $f(x)$ is $-1$ , find $a$ . 2. If $a = 1$ , prove that for $x \geqslant 0$ ,

f(x) + 2x - x\ln(x+1) \geqslant 0.

Answer

Solution

Difficulty	6/10
Topics	Extrema, calculus, Monotonicity, Logarithms, inequality

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