A Sum Bounded by One

★ 7/10functions Extrema calculus Monotonicity Casework

Problem

Let $f(x) = 2ax + \mathrm{e}^x$ and $g(x) = ax^2 - 2ax - x\mathrm{e}^x$ , with $a \in \mathbb{R}$ .

1. Discuss the monotonic intervals of $f$ . 2. If $f(x) + g(x) \le 1$ for all real $x$ , find the range of $a$ .

Answer

Solution

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

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