I need help with a question. It says:
Graph and state all important points and intervals: y=(lnx)/(x)
Well lets start by taking the derivative
$\displaystyle \frac{dy}{dx}=\frac{x \cdot \frac{1}{x}-(1) \cdot \ln(x)}{x^2}=\frac{1-\ln(x)}{x^2}$
So our critical numbers are the zero's of the derivative
$\displaystyle 1 -\ln(x)=0 \iff 1 =\ln(x) \iff e=x$
The inflection points are given by the 2nd derivative
$\displaystyle \frac{d^2y}{dx^2}=\frac{x^2 \cdot\frac{-1}{x}-(2x)(1-\ln(x)}{x^4}=\frac{-3x+\ln(x)}{x^4}$
setting equal to zero we get
$\displaystyle -3x+\ln(x)=0$
This has no Real solutions see graphs below
So the graph will look like the next one
Good luck