An Absolute-Log Barrier

★ 7/10functions Extrema calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = \left(2x^2 - x^3\right)\mathrm{e}^{1-x}$ for $x > 0$ .

1. Find the maximum value of $f(x)$ . 2. If the inequality

ax^2\mathrm{e}^{1-x} + |\ln x| \geqslant a

holds for every $x \in (0, +\infty)$ , find the range of possible values of the real number $a$ .

Answer

Solution

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

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