Linked by a Common Value

★ 7/10functions Extrema calculus Substitution Logarithms

Problem

Let $f(x) = \dfrac{\ln x}{x}$ and $g(x) = x\mathrm{e}^{-x}$ . Suppose there exist $x_1 \in (0, +\infty)$ and $x_2 \in \mathbb{R}$ with

f(x_1) = g(x_2) = k, \qquad k < 0.

Find the maximum value of

\left(\frac{x_2}{x_1}\right)^2 \mathrm{e}^k.

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, calculus, Substitution, Logarithms

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