A Sharp Numeric Bound

★ 6/10Extrema calculus Monotonicity Estimation

Problem

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

1. If $a = -1$ , find the intervals on which $f(x)$ is monotonic. 2. If $a \in [0, 1]$ , prove that

32 f(x) > -7.

(You may use the estimates $\ln 3 \approx 1.099$ and $\ln 4 \approx 1.386$ .)

Answer

Solution

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

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