The twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f ^ { \prime }$, $g$, and $g ^ { \prime }$ for various values of $x$ are given in the table below.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$x$ & -2 & $- 2 < x < - 1$ & -1 & $- 1 < x < 1$ & 1 & $1 < x < 3$ & 3 \\
\hline
$f ( x )$ & 12 & Positive & 8 & Positive & 2 & Positive & 7 \\
\hline
$f ^ { \prime } ( x )$ & -5 & Negative & 0 & Negative & 0 & Positive & $\frac { 1 } { 2 }$ \\
\hline
$g ( x )$ & -1 & Negative & 0 & Positive & 3 & Positive & 1 \\
\hline
$g ^ { \prime } ( x )$ & 2 & Positive & $\frac { 3 } { 2 }$ & Positive & 0 & Negative & -2 \\
\hline
\end{tabular}
\end{center}

(a) Find the $x$-coordinate of each relative minimum of $f$ on the interval $[ - 2, 3 ]$. Justify your answers.\\
(b) Explain why there must be a value $c$, for $- 1 < c < 1$, such that $f ^ { \prime \prime } ( c ) = 0$.\\
(c) The function $h$ is defined by $h ( x ) = \ln ( f ( x ) )$. Find $h ^ { \prime } ( 3 )$. Show the computations that lead to your answer.\\
(d) Evaluate $\displaystyle\int _ { - 2 } ^ { 3 } f ^ { \prime } ( g ( x ) ) g ^ { \prime } ( x ) \, dx$.