Question

If the graphs of the functions $y=\ln x$ and $y=ax$ intersect at exactly two points, then $a$ must be

• A. $(0,e)$
• B. $(\frac{1}{e},0)$
• C. $(0,\frac{1}{e})$
• D. None of these

Answer 1

Answer: (C)

$(0,\frac{1}{e})$

Given curves are $y=\ln x$ and $y=ax$.
$\Rightarrow \ln x=ax$ has exactly two solutions.
$\Rightarrow \frac{\ln x}{x}=a$ has exactly two solutions to find the range of $\frac{\ln x}{x}$.
Let $y=\frac{\ln x}{x},x>0$
$\frac{dy}{dx}=\frac{x.\frac{1}{x}-\ln x}{x^2}=\frac{1-\ln x}{x^2}$
$y$ is increasing, if $1-\ln x>0$ or $\ln x<1$
$\Rightarrow 0<x<e$
Range of $y\in (-\infty ,\frac{1}{e})$ graph of $y=\frac{\ln x}{x}$
For exactly two solutions of $\frac{\ln x}{x}=a$
$\Rightarrow a\in (0,\frac{1}{e})$