A Cosine Comparison Inequality

★ 8/10functions Extrema calculus Monotonicity Casework

Problem

Let $f(x) = \mathrm{e}^x - \mathrm{e}x$ and

h(x) = af(x) + 2f(-x) + (2a - 4)\mathrm{e}x \qquad (a \ne 0).

1. Discuss the monotonicity of $y = f(ax)$ . 2. If for all $x \ge 0$ ,

h(x) \ge (a+2)\cos x + (\mathrm{e}-1)(a-2)x,

find the range of $a$ .

Answer

Solution

Difficulty	8/10
Topics	functions, Extrema, calculus, Monotonicity, Casework

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