Dominating a Square Root

★ 7/10calculus Monotonicity Estimation inequality

Problem

Let $f(x) = \mathrm{e}^{x-a} - ax$ with $a \in \mathbb{R}$ .

1. If $f(x)$ has two zeros, find the range of possible values of $a$ . 2. If for every $x \in [0, +\infty)$ ,

f(x+1) + \frac{a}{2}(x+2) \geqslant \sqrt{x^2 + ax + 1},

find the range of possible values of $a$ .

Answer

Solution

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

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