Question

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

• A. $\frac{e}{3}$
• B. $3e$
• C. $\frac{1}{3e}$
• D. $\frac{3}{e}$

Answer 1

The correct option is C $\frac{1}{3e}$

$y^{{}^{\prime }}=\frac{1-\ln x}{x^2}$ and $y^{{}^{\prime }}=2\lambda x$
$1-\ln x=2\lambda x^3$
$1-\ln x=2\ln x(\because \lambda x^3=\ln x)$
$\ln x=\frac{1}{3}\Rightarrow x=e^{\frac{1}{3}}$

Putting the value of $x$ in $\lambda x^3=\ln x$
$\frac{1}{3}=\lambda e\Rightarrow \lambda =\frac{1}{3e}$