Blog: `y = x/ln(x)` Locate any relative extrema and points of inflection.

Find the extrema and points of inflection for the graph of $\displaystyle{y}=\frac{{x}}{{{\ln{{x}}}}}$ :

Extrema can only occur at critical points, or where the first derivative is zero or fails to exist.

Note that the domain for the function is x>0, $\displaystyle{x}\ne{1}$ .

$\displaystyle{y}'=\frac{{{\ln{{x}}}-{1}}}{{{\left({\ln{{x}}}\right)}}^{{2}}}$ This function is defined for all x in the domain so we set it equal to zero. Note that a fraction is zero if the numerator,...

Find the extrema and points of inflection for the graph of $\displaystyle{y}=\frac{{x}}{{{\ln{{x}}}}}$ :

Extrema can only occur at critical points, or where the first derivative is zero or fails to exist.

Note that the domain for the function is x>0, $\displaystyle{x}\ne{1}$ .

$\displaystyle{\ln{{x}}}-{1}={0}=\Rightarrow{\ln{{x}}}={1}=\Rightarrow{x}={e}$ . For 1<x<e the first derivative is negative and for x>e it is positive, so the only extrema is a minimum at x=e.

Any inflection point can only occur if the second derivative is equal to zero.

$\displaystyle{y}''=\frac{{\frac{{{{\ln}}^{{2}}{x}}}{{x}}+\frac{{{2}{\ln{{x}}}}}{{x}}}}{{{{\ln}}^{{4}}{x}}}$ or

$\displaystyle{y}''=\frac{{{2}+{\ln{{x}}}}}{{{x}{{\ln}}^{{3}}{x}}}$ Which is negative for x<1, and positive for x>1. The graph is concave down on 0<x<1 and concave up on x>1, but x=1 is undefined so there are no inflection points.