A Double Quantifier Condition

★ 8/10functions Extrema calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = e^x - ax$ and $g(x) = \ln x - ax$ . Suppose that for every $x_1 \in (0, +\infty)$ there exists $x_2 \in (0, +\infty)$ such that

f(x_1)\,g(x_2) = x_1 x_2.

Find the range of possible values of the real number $a$ .

Answer

Solution

Difficulty	8/10
Topics	functions, Extrema, calculus, Monotonicity, Logarithms, inequality

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