A Derivative First

★ 6/10Extrema calculus Monotonicity Estimation inequality

Problem

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

1. Let $f'(x)$ be the derivative of $f(x)$ . Discuss the monotonicity of $y = f'(x)$ . 2. When $a \leqslant 1 - \dfrac{1}{\mathrm{e}}$ , prove that

f(x) + x - \ln(x+1) \geqslant 1.

Answer

Solution

Difficulty	6/10
Topics	Extrema, calculus, Monotonicity, Estimation, inequality

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