An Extremum and a Hard Bound

★ 7/10functions Extrema calculus Substitution Monotonicity

Problem

Let $f(x) = a\mathrm{e}^x - \sin x$ , where $a$ is a real number.

1. If $f(x)$ has an extreme value on the interval $\left(0, \dfrac{\pi}{2}\right)$ , find the range of possible values of $a$ . 2. If the inequality

f(x) + \sin x - 1 \leqslant x\mathrm{e}^x\left(x + \ln x - 1\right)

holds for all $x$ in its domain, find the range of possible values of $a$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, calculus, Substitution, Monotonicity

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