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

Find the extrema and points of inflection for the graph of y=x/(lnx) :
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, x ne 1 .
y'=(lnx-1)/(lnx)^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, but not the denominator, is equal to zero.
lnx-1=0 ==> lnx=1 ==> x=e . For 1e 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.
y''=((ln^2x)/x+(2lnx)/x)/(ln^4x) or
y''=(2+lnx)/(xln^3x) Which is negative for x<1, and positive for x>1. The graph is concave down on 01, but x=1 is undefined so there are no inflection points.
The graph: