1. The problem states that for the function $g(x)$, we have $g'(2) = 0$ and $g''(2) = 5$. 2. To classify whether $x = 2$ is a local minimum, maximum, or an inflection point, we use the second derivative test. 3. The second derivative test states: if $g'(a) = 0$ and $g''(a) > 0$, then $g(x)$ has a local minimum at $x = a$. 4. Here, $g'(2) = 0$, and $g''(2) = 5 > 0$, so the function has a local minimum at $x = 2$. 5. Thus the correct answer is option A) Local minimum.