An Exponential Ceiling

★ 8/10Extrema calculus Monotonicity inequality

Problem

Let

f(x) = -\frac{a}{2}x^2 + (a-1)x + \ln x.

1. If $a = -\dfrac{1}{2}$ , find the intervals of monotonicity of $f(x)$ . 2. If $a > 1$ , prove that

(2a - 1)f(x) < 3e^{a-3} \qquad \text{for all } x > 0.

Answer

Solution

Difficulty	8/10
Topics	Extrema, calculus, Monotonicity, inequality

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