An Exponential-Log Product

★ 7/10calculus Tangent Line Trick Monotonicity Logarithms inequality

Problem

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}x^2 - ax - 1$ with $a \in \mathbb{R}$ .

1. If $f(x) \geqslant 0$ for all $x \in [0, +\infty)$ , find the range of possible values of $a$ . 2. Prove that for $x > 0$ ,

\left(\mathrm{e}^x - \frac{1}{2}x^2 + 1\right)\ln(x+1) > 2x.

Answer

Solution

Difficulty	7/10
Topics	calculus, Tangent Line Trick, Monotonicity, Logarithms, inequality

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