Question

If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ constant) touch each other, then $3e\lambda$ is

• A. $e$
• B. $e^2$
• C. 1
• D. 9

Answer 1

The correct option is C 1

$y^{\prime }=\frac{1-\ln x}{x^2}\text{and}{y}^{\prime }=2\lambda x$
$\begin{array}{c}\Rightarrow 1-\ln x=2\lambda x^3\\ \ln x=\lambda x^3\end{array}\}$
On solving these two equations simultaneously we get,
$\ln x=\frac{1}{3}\Rightarrow x=e^{\frac{1}{3}}$
After equating $y$ values of both curves for the calculated $x$ we get,
$\frac{1}{3}=\lambda e\Rightarrow \lambda =\frac{1}{3e}$