Activity 3.3.2.
Suppose that \(g(x)\) is a function whose first derivative is
\begin{equation*}
g'(x) = \frac{(x+4)(x-2)}{x^2+1}\text{.}
\end{equation*}
(a)
Determine, with justification, all critical numbers of \(g\text{.}\)
(b)
By developing a carefully labeled first derivative sign chart, decide whether \(g\) has as a local maximum, local minimum, or neither at each critical number.
(c)
Does \(g\) have a global maximum? global minimum? Justify your claims.
(d)
Sketch a possible graph of \(y = g(x)\text{.}\) Clearly label any local or global extrema on the graph.
