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 0<x<1 the first derivative is negative, and for x>1 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:
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