A Widening Gap

★ 8/10functions Extrema calculus Convexity Monotonicity

Problem

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}ax^2 - x$ .

1. If $f(x)$ is increasing on all of $\mathbb{R}$ , find the value of $a$ . 2. For $a > 1$ :

Prove that $f(x)$ has two extreme points $x_1 < x_2$ , and that $x_2 - x_1$ increases as $a$ increases.
Under the conclusion of part 1, prove that

f(x_2) < 1 + \frac{\sin x_2 - x_2}{2}.

Answer

Solution

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

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