y = x^2/2 - ln(x) Locate any relative extrema and points of inflection.

We are asked to locate the relative extema and any inflection points for the graph of y=x^2/2-lnx :
Extrema can occur only at critical points; i.e. points in the domain where the first derivative is zero or fails to exist. So we find the first derivative:
y'=x-1/x   This is a continuous and differentiable function everywhere except x=0, which is not in the domain of the original function. (The domain, assuming real values, is x>0.)
Setting the first derivative equal to zero we get:
x-1/x=0 ==> x=1/x ==> x^2=1 ==> x=1   (x=-1 is not in the domain.)
For 01 it is positive, so there is a minimum at x=1. This is the only minimum or maximum.
Inflection points can only occur when the second derivative is zero:
y''=1+1/x^2 ==> y''>0 forall x so there are no inflection points.
The graph: