A Log-Shift Refinement

★ 9/10Extrema calculus Monotonicity inequality

Problem

Let $f(x) = (x-1)e^x - ax + 1$ .

1. For $a = 0$ , solve the inequality $f(x) \leqslant 0$ . 2. Prove that for every $a > 0$ ,

f\left(\ln(1 + a)\right) < 0.

Restrict $f$ to the domain $\left(t, \ln\left(e^t + a\right)\right)$ and let $A$ be its range. Find a necessary and sufficient condition (on $t$ ) for $A \subseteq (-\infty, 0)$ to hold for every $a > 0$ .

Answer

Solution

Difficulty	9/10
Topics	Extrema, calculus, Monotonicity, inequality

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