Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table below, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & $0 < x < 1$ & 1 & $1 < x < 2$ & 2 & $2 < x < 3$ & 3 & $3 < x < 4$ \\
\hline
$f ( x )$ & - 1 & Negative & 0 & Positive & 2 & Positive & 0 & Negative \\
\hline
$f ^ { \prime } ( x )$ & 4 & Positive & 0 & Positive & DNE & Negative & - 3 & Negative \\
\hline
$f ^ { \prime \prime } ( x )$ & - 2 & Negative & 0 & Positive & DNE & Negative & 0 & Positive \\
\hline
\end{tabular}
\end{center}

(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.\\
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.\\
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.\\
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.