Two Extremes of an Exponential

★ 7/10Extrema calculus Monotonicity inequality

Problem

The function $f(x) = \mathrm{e}^{2x} - ax^2 - 1$ (for $x \in \mathbb{R}$ ) has two extreme points $x_1, x_2$ .

1. Find the range of possible values of the real number $a$ . 2. Prove that

a\mathrm{e}^{2x_1} + a\mathrm{e}^{2x_2} > 2\mathrm{e}^{2x_1}\cdot\mathrm{e}^{2x_2}.

Answer

Solution

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

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