A Classic Exponential-Log Bound

★ 7/10functions Extrema calculus Tangent Line Trick Monotonicity

Problem

Let $f(x) = \mathrm{e}^x - \ln(x + m)$ .

1. Suppose $x = 0$ is an extreme point of $f(x)$ . Find $m$ , and discuss the monotonicity of $f(x)$ . 2. Prove that if $m \leqslant 2$ , then $f(x) > 0$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, calculus, Tangent Line Trick, Monotonicity

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