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Let $g$ be a function defined for all $x>0$ such that $g(x)=\frac{1}{2}{x}^2-2lnx.$

a$)$ Find $g^{\text{'}}(x).$

b$)$ Find all values of $x$ for which the graph of $g$ has a borizontal tangent, and determine whether $g$ has a local maximum, a local minimum, or neither at each of these values. Justify your answer.

c$)$ On what intervals, if any, is the graph of $g$ concave up$?$ Justify your answer.