Equal Ratios of Logs

★ 7/10functions calculus Tangent Line Trick Monotonicity Logarithms inequality

Problem

Let $f(x) = \ln x$ and $g(x) = \mathrm{e}^x$ .

1. Find the intervals on which $y = f(x) - x$ is monotonic. 2. Prove that on the common domain of $f$ and $g$ ,

g(x) - f(x) > 2.

If there exist two distinct real numbers $x_1, x_2$ with

\frac{f(x_1)}{x_1} = \frac{f(x_2)}{x_2} = a,

prove that $\dfrac{x_1 x_2}{\mathrm{e}^2} > 1$ .

Answer

Solution

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

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